Number of Roots of a Quadratic
Read this topicExplanations 8
- The easiest way to think about a quadratic equation with no real solutions is to first visualize the graph. First, look at this graph: Using a simple process of elimination, we …
By Nisha Devasia, Alex Federspiel, Sarah Walker
- When the discriminant (b2−4ac)(b^2 - 4ac)(b2−4ac) is positive, square roots of positive numbers are always real, so the solution(s) will be real. However, there can be one solut…
By Vivian Loh, Moor Xu, Alex Federspiel, Victor Wang
- Consider a quadratic ax2+bx+cax^2+bx+cax2+bx+c. We should knock off the easiest part of this question first. The yyy-intercept occurs when x=0x=0x=0. Plugging x=0x=0x=0 into you…
By Wayne Zhao, Moor Xu, Sarah Walker, Alex Federspiel
- The nature of the roots of a quadratic equation can be determined by taking the discriminant of the equation. By nature, it is meant as whether the roots are real distinct, real…
By Sharky Kesa, Ben Ferrara, Alex Federspiel
- When do Quadratic Equations have real solutions? Recall that a quadratic equation, written in the form ax2+bx+cax^2 +bx + cax2+bx+c, has the solutions −b±b2−4ac2a\frac{-b \pm \s…
By Tommy San Miguel, Alex Federspiel, Sarah Walker, Amy Feraco
- (Video) Factoring and the Quadratic Formula by PatrickJMT Patrick JMT has a great video explaining why there are quadratics with no real solutions and what to do about them. Sum…
By Ben Ferrara, Alex Federspiel, Sarah Walker, Amy Feraco
- (Video) Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation by Patrick JMT In this video, PatrickJMT states that we will be learning how to predict …
By Ben Ferrara, Alex Federspiel, Sarah Walker, Amy Feraco
- (Video) The Discriminant and Quadratic Graphs by HEGARTYMATHS Here's a video by HEGARTYMATHS showing how the discriminant relates to the number of xxx-intercepts the graph will …
By Alex Federspiel, Amy Feraco
Problems 77
- Which of the following quadratic equations have real solutions? Select all the correct answers.
- Identify if there are two roots, one root, or no roots in x2−20x−51x^2 - 20x - 51x2−20x−51.
- How many real solutions does the equation 3x2−4x−2=03x^2 - 4x - 2=03x2−4x−2=0 have?
- Suppose that a quadratic function has discriminant −3-3−3. How many real roots does it have?
- A quadratic polynomial has roots 444 and −3-3−3. What do we know about the discriminant Δ\DeltaΔ? Check all that apply.
- Let fff be the quadratic function with leading coefficient 111 whose graph is tangent to that of the lines y=−5x+6y=-5x+6y=−5x+6 and y=x−1y=x-1y=x−1. The sum of the coefficients…
- What is the discriminant of the quadratic −3x2+2x+1-3x^2+2x+1−3x2+2x+1? Does this mean that the roots are real or non-real?
- What is the discriminant of the quadratic −12x2+3x−17-\frac{1}{2}x^2+3x-17−21x2+3x−17? Does this mean that the roots are real or non-real?
- What is the discriminant of the quadratic 13x2+4x+3\frac{1}{3}x^2+4x+331x2+4x+3? Does this mean that the roots are real or non-real?
- Which parabola(s) above is/are the graph of a quadratic with no real roots?
- Which parabola(s) above is/are the graph of a quadratic with no real roots?
- True or False\text{True or False}True or False === If the discriminant is 000, then all roots are non-real.
- Which function y(x)y(x)y(x) below has non-real roots?
- A quadratic polynomial has exactly one root, −2-2−2. What do we know about the discriminant Δ\DeltaΔ? Check all that apply.
- Suppose that a quadratic function has discriminant 400400400. How many roots does it have?
- Find the discriminant of the quadratic y(x)=2x2−5x+3y(x)=2x^2-5x+3y(x)=2x2−5x+3, and call it Δ\DeltaΔ. How many real solutions does y(x)=0y(x) = 0y(x)=0 have?
- Find the discriminant of y(x)=12x2+6x+21y(x)=\frac{1}{2}x^2+6x+21y(x)=21x2+6x+21, and call it Δ.\Delta.Δ. How many real solutions does y(x)=0y(x) = 0y(x)=0 have?
- Find the discriminant of y(x)=−4.5x2+12x−8y(x)=-4.5x^2+12x-8y(x)=−4.5x2+12x−8. How many real solutions does y(x)=0y(x) = 0y(x)=0 have?
- Identify if there are two roots, one root, or no roots in x2=0x^2 = 0x2=0.
- Identify if there are two roots, one root, or no roots in x2−x−2x^2 - x - 2x2−x−2.
- Identify if there are two roots, one root, or no roots in x2+x+6x^2 + x + 6x2+x+6.
- Identify if there are 222 roots, 111 root, or no roots for y=−4x2+8x−4y = -4x^2 + 8x - 4y=−4x2+8x−4.
- Identify if there are 222 roots, 111 root, or no roots in 3x2−4x+23x^2 - 4x + 23x2−4x+2.
- Identify if there are two roots, one root, or no roots in 2x2−25x+12x^2-25x+12x2−25x+1.
- If the quadratic x2+bx+cx^2 + bx + cx2+bx+c has c<0c < 0c<0, how many roots does it have?
- If f(x)=ax2+bx+cf(x)=ax^2+bx+cf(x)=ax2+bx+c is a quadratic function with a>0a>0a>0 that has no solutions, what must be true of the vertex (x,y)(x,y)(x,y)?
- If f(x)=ax2+bx+cf(x)=ax^2+bx+cf(x)=ax2+bx+c is a quadratic function with a<0a<0a<0 that has no solutions, what must be true of the vertex (x,y)(x,y)(x,y)?
- If f(x)=ax2+bx+cf(x)=ax^2+bx+cf(x)=ax2+bx+c is a quadratic function with a>0a>0a>0 that has two real solutions, what must be true of the vertex (x,y)(x,y)(x,y)?
- If f(x)=ax2+bx+cf(x)=ax^2+bx+cf(x)=ax2+bx+c is a quadratic function with a<0a<0a<0 that has two real solutions, what must be true of the vertex (x,y)(x,y)(x,y)?
- Select all options that make the following sentence true: If the discriminant is ‾\underline{\ \ \ \ \ \ \ \ \ \ \ } then there are ‾\underline{\ \ \ \ } real solutions. \\ …
- Select all options that make the following sentence true: If the discriminant is ‾\underline{\ \ \ \ \ \ \ \ \ \ \ } then there are ‾\underline{\ \ \ \ } real solutions. \\ …
- Select all options that make the following sentence true: If the discriminant is ‾\underline{\ \ \ \ \ \ \ \ \ \ \ } then there are ‾\underline{\ \ \ \ } real solutions. \\ …
- If a quadratic equation of the form x2+bx+c=0x^2+bx+c=0x2+bx+c=0 has two real solutions, which of the following must be true of ccc?
- If a quadratic equation of the form x2+c=0x^2+c=0x2+c=0 has two real solutions, which of the following must be true of ccc?
- If a quadratic equation of the form ax2+c=0ax^2+c=0ax2+c=0 has one real solution, which of the following must be true of ccc? Assume a≠0.a \neq 0.a=0.
- If a quadratic equation of the form −x2+bx+c=0-x^2+bx+c=0−x2+bx+c=0 has two real solutions, which of the following must be true of ccc?
- For a general quadratic equation fff, which two of the following are equal?
- Which of the following graphed quadratic equations has a positive discriminant?
- Which of the following graphed quadratic equations has a negative discriminant?
- Which of the following graphed quadratic equations has a zero discriminant?
- If a quadratic equation of the form x2+bx+c=0x^2+bx+c=0x2+bx+c=0 has two real solutions, which of the following must be true of ccc?
- If a quadratic equation of the form x2+c=0x^2+c=0x2+c=0 has two real solutions, which of the following must be true of ccc?
- If a quadratic equation of the form −x2+bx+c=0-x^2+bx+c=0−x2+bx+c=0 has two real solutions, which of the following must be true of ccc?
- If a quadratic equation f(x)f(x)f(x) has a positive (nonzero) determinant, which of the following could be the graph of this function?
- Which of the following graphs does not have any real solutions?
- Which of the following are real-numbered solutions? (Select all that apply.)
- If the discriminant of a quadratic equation Δ=100\Delta=100Δ=100, how many real solutions will there be?
- If the discriminant of a quadratic equation Δ=14\Delta=\frac{1}{4}Δ=41, how many real solutions will there be?
- If the discriminant of a quadratic equation Δ=16π2\Delta=16\pi^2Δ=16π2, how many real solutions will there be?
- Concept check: If a quadratic equation has two real-numbered solutions, which of the following must be true of the discriminant Δ\DeltaΔ?
- For which values of aaa will the equation ax2+4x−1=0ax^2+4x-1=0ax2+4x−1=0 have at least one real solution?
- For which values of ccc will the equation 2x2+4x+c=02x^2+4x+c=02x2+4x+c=0 have at least one real solution?
- Which of the following are real-numbered solutions? (Select all that apply.)
- If the discriminant of a quadratic equation is Δ=−14\Delta=-\frac{1}{4}Δ=−41, how many real solutions will there be?
- If the discriminant of a quadratic equation Δ=−100\Delta=-100Δ=−100, how many real solutions will there be?
- If the discriminant of a quadratic equation is Δ=−0.0000000001\Delta=-0.0000000001Δ=−0.0000000001, how many real solutions will there be?
- For which values of aaa will the equation ax2+4x−1=0ax^2+4x-1=0ax2+4x−1=0 have no real solutions?
- For which values of ccc will the equation 2x2+4x+c=02x^2+4x+c=02x2+4x+c=0 have no real solutions?
- If a quadratic equation of the form x2+bx+c=0x^2+bx+c=0x2+bx+c=0 has no real solutions, which of the following must be true of ccc? (Select all that apply.)
- When does a quadratic equation of the form x2+c=0x^2+c=0x2+c=0 have no real solutions?
- If a quadratic equation f(x)f(x)f(x) has a negative determinant, which of the following could be the graph of this function?
- Concept check: If a quadratic equation has zero real-numbered solutions, which of the following must be true of the discriminant Δ\DeltaΔ?
- Which of the following graphs do not have any real solutions?
- Find the smallest positive integer ccc such that 4x2−cx+94x^2 - cx + 94x2−cx+9 factors into a product of two, first degree polynomials with real coefficients.
- D=b2−4acD = b^2 - 4acD=b2−4ac If D>0D > 0D>0, which of the following statements is true?
- D=b2−4acD = b^2 - 4acD=b2−4ac If D=0D = 0D=0, which of the following statements is true?
- D=b2−4acD = b^2 - 4acD=b2−4ac If D<0D < 0D<0, which of the following statements is true?
- D=b2−4acD = b^2 - 4acD=b2−4ac If b2=4acb^2 = 4acb2=4ac, which of the following statements is true?
- D=b2−4acD = b^2 - 4acD=b2−4ac If b2>4acb^2 > 4acb2>4ac, which of the following statements is true?
- D=b2−4acD = b^2 - 4acD=b2−4ac If b2<4acb^2 < 4acb2<4ac, which of the following statements is true?
- Consider the equations below: D=b2−4acx=−b±D2a\begin{array}{rcl} D &=& b^2 - 4ac \\ x &=& \frac{-b \pm \sqrt{D}}{2a} \end{array}Dx==b2−4ac2a−b±D If D>0D > 0D>0, how many po…
- Consider the equation below: y=ax2+bx+c\begin{array}{rcl} y &=& ax^2 + bx + c \\ \end{array}y=ax2+bx+c If b2>4acb^2 > 4acb2>4ac, how many real roots does yyy have?
- Consider the equation below: y=ax2+bx+c\begin{array}{rcl} y &=& ax^2 + bx + c \\ \end{array}y=ax2+bx+c If b2=4acb^2 = 4acb2=4ac, how many real roots does yyy have?
- Consider the equations below: D=b2−4acx=−b±D2a\begin{array}{rcl} D &=& b^2 - 4ac \\ x &=& \frac{-b \pm \sqrt{D}}{2a} \end{array}Dx==b2−4ac2a−b±D If D=0D = 0D=0, how many po…
- Consider the given statements below: y=ax2+bx+cb2=4ac\begin{array}{rcl} y &=& ax^2 + bx + c \\ b^2 &=& 4ac \end{array}yb2==ax2+bx+c4ac If a,b,ca,b,ca,b,c are constants, which…
- Consider the given statements below: y=ax2+bx+cb2−4ac=kk>0\begin{array}{rcl} y &=& ax^2 + bx + c \\ b^2 - 4ac&=& k \\ k &>& 0 \end{array}yb2−4ack==>ax2+bx+ck0 If a,b,ca,b,ca,…
- Let fff and ggg be polynomials with degree at most 222 that have the same discriminant and the same linear coefficient. If ggg has 000 as an xxx-intercept, then which of the fol…