Discriminant to Find Nature of Roots
Read this topicExplanations 7
- Now that we know what quadratics are, a pressing question arises. Just as we were able to find solutions to linear equations like 2x+3=52x + 3 = 52x+3=5, can we find the solutio…
By Vinjai Vale, Victor Wang, Rajiv Movva, Expii Team, Anusha Rahman
- Have you ever wondered what type of solutions a quadratic equation will have without actually finding the solutions? The discriminant does just that! It is the part of the quadr…
By Mr. Richardson, Jordan Safer, Moor Xu
- Let our equation be of the form x2+bx+c=0x^2+bx+c=0x2+bx+c=0 and let x1x_1x1 and x2x_2x2 be the roots of the quadratic. Then x2+bx+c=(x−x1)(x−x2)x^2+bx+c=(x-x_1)(x-x_2)x2+bx+c…
By Michael Livshits, Anusha Rahman
- What is the Quadratic Formula? When given an equation in the form ax2+bx+cax^2 + bx + cax2+bx+c, usually the simplest way to solve for the value of xxx is to factor the quadrati…
By Sarah Walker, Victor Wang
- What is a Complex Root? Now that you have had some practice with the quadratic formula, its time to move on to complex roots. This can sound scary at first, but I promise that o…
By Sarah Walker, Victor Wang
- The Discriminant The discriminant of a quadratic equation tells you the nature of the roots. That means you can use the discriminant to find out whether a quadratic function has…
By Anusha Rahman
- (Video) Using the Discriminant to Find How Many Roots by FerranteMath In this video by FerranteMath, you will learn how to determine the number of roots in a quadratic by using …
By Anusha Rahman
Problems 32
- How many solutions, and what type are they, are there to the equation 4x2−8x+4=04x^2-8x+4 = 04x2−8x+4=0?
- Find the value of 12+12+12⋯\sqrt{12 + \sqrt{12 + \sqrt{12 \cdots }}}12+12+12⋯ .
- Find the discriminant of the quadratic x2−12x+4x^2 - 12x + 4x2−12x+4, and use it to classify the solutions of x2−12x+4=0x^2 - 12 x + 4 = 0x2−12x+4=0.
- Classify the solutions of the quadratic equation x2+5x+8=0x^2 + 5x + 8 = 0x2+5x+8=0.
- Solve for the possible integer solution(s) for this equation. x2+xy+y2=(x+y3+1)3.x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.x2+xy+y2=(3x+y+1)3.
- Find all solutions to (m2+n)(m+n2)=(m−n)3(m^2+n)(m + n^2)= (m - n)^3(m2+n)(m+n2)=(m−n)3, where mmm and nnn are non-zero integers.
- Find all integer solutions (x,y)(x,y)(x,y) to 7x2−13xy+7y23=∣x−y∣+1.\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.37x2−13xy+7y2=∣x−y∣+1.
- How many solutions, and what type are they, are there to the equation x2−6x+8x^2-6x+8x2−6x+8?
- How many solutions, and what type are they, are there to the equation x2−9x+4x^2-9x+4x2−9x+4
- How many solutions, and what type are they, are there to the equation 2x2−9x+112x^2-9x+112x2−9x+11
- If f(x)=2x2+2x+1f(x)=2x^2+2x+1f(x)=2x2+2x+1, what is the sum of the two roots of f(x)=0f(x)=0f(x)=0?
- What are the roots of the quadratic equation 3x2+4x+2=03x^2+4x+2=03x2+4x+2=0? Hint: use the quadratic equation (x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac)
- Let a,b,ca,b,ca,b,c be real numbers and f(x)=ax2+bx+cf(x)=ax^2+bx+cf(x)=ax2+bx+c be a polynomial. Suppose that for any non-real complex number xxx, the value f(x)f(x)f(x) is als…
- Find xxx: 2x2+xx3=2\frac{2x^2+x}{x^3}=2x32x2+x=2
- How many real solutions does the equation x2+6x+10=0x^2+6x+10=0x2+6x+10=0 have?
- Solve for xxx in 3x2+9x+4=03x^2+ 9x + 4 = 03x2+9x+4=0.
- Solve for xxx in 13x=5x2+813x = 5x^2 + 813x=5x2+8.
- Solve for xxx in 2x(x+3)=42x(x+3) = 42x(x+3)=4.
- Solve for xxx in x2+4x=7\frac{x^2+4}{x}=7xx2+4=7.
- Solve for the possible values of xxx. If you get a fractional answer, round down to the nearest whole number. (Hint: there may be more than one correct answer!)
- A marching band is looking for a new section of lead trumpets. However, in order to get the best sound they need a certain number of players. The following equation was calculat…
- Solve for xxx using the quadratic formula: x2+3x+56=0x^{2}+3x+\dfrac{5}{6} = 0x2+3x+65=0
- Solve for xxx using the quadratic formula: x2=20−4xx^{2}= 20-4xx2=20−4x
- Solve for xxx using the quadratic formula: 3x(3x+1)=23x(3x+1)=23x(3x+1)=2
- Solve for xxx using the quadratic formula: 4x2=−x+54x^{2}=-x+54x2=−x+5
- Solve for xxx using the quadratic formula: x2−8x+14=0x^{2}-8x+14=0x2−8x+14=0
- Solve for xxx using the quadratic formula: 16x2+43x−23=0\dfrac{1}{6}x^{2}+\dfrac{4}{3}x-\dfrac{2}{3}=061x2+34x−32=0
- Solve for xxx using the quadratic formula: 4x2−6x=144x^{2} -6x=144x2−6x=14
- Solve for xxx using the quadratic formula: x=3x2−5x= 3x^{2}-5x=3x2−5
- Solve for xxx using the quadratic formula: x2+x−1=0x^{2}+x-1=0x2+x−1=0
- Can you use the quadratic equation to solve the following expression? 32x2+0.37x−1231=0\frac{3}{2}x^2 + 0.37x - \frac{12}{31} = 023x2+0.37x−3112=0
- Can you use the quadratic equation to solve the following expression? 4x+3=04x + 3 = 04x+3=0