How to Solve Quadratic Equations
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- Нехай за умовою дано квадратне рівняння ax2+bx+с=0ax^2 + bx + с = 0ax2+bx+с=0. Розв'язати таке квадратне рівняння означає знайти всі числа xxx такі, що рівність стає правильною.…
By Sasha Rudenko, Nikita Skybytskyi · Українська
Problems 38
- Solve for xxx. x2+8=22x−1\sqrt{x^2+8}=2\sqrt{2x-1}x2+8=22x−1
- Solve for xxx. x−6=7x−22\sqrt{x-6}=\sqrt{7x}-22x−6=7x−22
- Solve 5x+6+3=3x+3+4\sqrt{5x+6}+3=\sqrt{3x+3}+45x+6+3=3x+3+4.
- Solve x−3=4−2x+1\sqrt{x-3}=4-\sqrt{2x+1}x−3=4−2x+1.
- Solve −2x−6=x+3\sqrt{-2x-6}=x+3−2x−6=x+3.
- Joe solves x−3=30−2xx-3=\sqrt{30-2x}x−3=30−2x by squaring to get (x−3)2=30−2x(x-3)^2 = 30-2x(x−3)2=30−2x, and solving the resulting quadratic. Which of the solutions to the qua…
- Find the roots of the quadratic function y(x)=−2x2+7x−6.y(x)=-2x^2+7x-6.y(x)=−2x2+7x−6. Try to factor, but if that isn't working or is too hard, remember the quadratic formula i…
- Find the zeros of 0=−3x2+3x+60=-3x^2+3x+60=−3x2+3x+6.
- Find the roots of 3x2+4x+1=0.3x^2+4x+1=0.3x2+4x+1=0. Try to factor, but if that isn't working or is too hard, remember the quadratic formula is x=−b±b2−4ac2a.x=\frac{-b\pm\sqrt{…
- CONCEPT CHECK: The table above shows a quadratic function y(x)y(x)y(x). Where approximately are the roots?
- What is the sum of the positive real solutions to (log2x)2=log2(x4)(\log_2 x)^2 = \log_2(x^4)(log2x)2=log2(x4)?
- True or False\text{True or False}True or False === If a quadratic equation is in the form x2+c=0x^2+c = 0x2+c=0 where c>0c>0c>0, then there are two real solutions for xxx.
- Find all solutions to the equation ∣x2−6x+7∣=2|x^2-6x+7| = 2∣x2−6x+7∣=2.
- Solve ∣x∣2−∣x∣−6=0|x|^2 - |x| - 6 = 0∣x∣2−∣x∣−6=0 for all real values of xxx.
- List all solutions of the equation 18x2−9−3x−3=1.\frac{18}{x^2-9} - \frac{3}{x-3} = 1.x2−918−x−33=1.
- It turns out 2+3=a+bc\sqrt{2 + \sqrt{3}} = \frac{a + \sqrt{b}}{\sqrt{c}}2+3=ca+b, where a,b,ca,b,ca,b,c are positive integers with no common factor, and b,cb,cb,c are "squa…
- Find the domain of f(x)=x−3x2+9x−22f(x)=\frac{x-3}{x^2+9x-22}f(x)=x2+9x−22x−3.
- Find all roots of the equation x2+2x+1=0.x^2+2x+1 = 0.x2+2x+1=0.
- Find the roots of 4x2+5x+1=04x^2+5x+1 = 04x2+5x+1=0. Mark all that apply.
- Find the roots of x2+7x+12=0.x^2+7x+12 = 0.x2+7x+12=0.
- Find the sum of the real roots of x2+17x+30=0.x^2+17x+30 = 0.x2+17x+30=0.
- Find the sum of the real solutions to the equation x2+12x+27=0.x^2+12x+27= 0.x2+12x+27=0.
- Find the sum of the real roots of x2+90x+729=0.x^2+90x+729 = 0.x2+90x+729=0.
- Find the equation of a quadratic function that has zeros at x=−2x=-2x=−2 and x=3x=3x=3.
- Find the equation of a quadratic function that has zeros at x=2x=2x=2 and x=5x=5x=5.
- Given the graph above, which of the following xxx-values are solutions to the equation f(x)=0f(x)=0f(x)=0? (Select all that apply.)
- Given the graph above, which of the following xxx-values are solutions to the equation f(x)=0f(x)=0f(x)=0? (Select all that apply.)
- Given the graph above, which of the following xxx-values are solutions to the equation f(x)=0f(x)=0f(x)=0? (Select all that apply.)
- Which of the following methods always work to determine whether or not a value xxx is a zero of a quadratic function fff? (Select all that apply.)
- Concept check: What does it mean for a value xxx to be a "zero" (or root) of a function fff? (Select all that apply.)
- There is a unique positive real number xxx such that x+0.01x2=1x + 0.01x^2=1x+0.01x2=1. What is xxx, rounded to the nearest whole number?
- The red line is y=x+1y = x + 1y=x+1 and the blue line is y=x−1y = x - 1y=x−1. If you only have this information, it would be best to find the parabola's roots with which of the …
- What would be the method that would be best for finding the roots of yyy?
- If y=3x2+22x+7,y = 3x^2 + 22x + 7 ,y=3x2+22x+7, what would be the least reliable method for solving y=0y=0y=0?
- If y=x2+5x+6,y = x^2 + 5x + 6,y=x2+5x+6, what would be the best method for finding the roots of yyy?
- Suppose x2+2kx=k2.x^2 + 2kx = k^2.x2+2kx=k2. What would be the best method to use in finding possible values of xxx? kkk is a constant.
- x2+(2n)x=k−n2x^2 + (2n)x = k - n^2x2+(2n)x=k−n2 nnn and kkk are constants. What are the possible values of xxx?
- A square has an area of 562556255625 mm2.^2.2. What is the length of its sides?