Question

Solve the following quadratic by completing the square. y = x^2 - 6x+7

Answer 1

Answer:

$x= 3+\sqrt{2}, \quad x= 3 -\sqrt{2}$

Step-by-step explanation:

Given quadratic equation:

$y = x^2 - 6x+7$

To complete the square, begin by adding and subtracting the square of half the coefficient of the term in x:

$\implies y = x^2 - 6x+\left(\dfrac{-6}{2}\right)^2-\left(\dfrac{-6}{2}\right)^2+7$

$\implies y = x^2 - 6x+\left(-3\right)^2-\left(-3\right)^2+7$

$\implies y = x^2 - 6x+9-9+7$

Factor the perfect square trinomial:

$\implies y=(x-3)^2-9+7$

$\implies y=(x-3)^2-2$

To solve the quadratic, set it to zero and solve for x:

$\implies (x-3)^2-2=0$

$\implies (x-3)^2-2+2=0+2$

$\implies (x-3)^2=2$

$\implies \sqrt{(x-3)^2}=\sqrt{2}$

$\implies x-3= \pm\sqrt{2}$

$\implies x-3+3= 3\pm\sqrt{2}$

$\implies x= 3\pm\sqrt{2}$

Therefore, the solution to the given quadratic equation is:

$x= 3+\sqrt{2}, \quad x= 3 -\sqrt{2}$

Answer 2

Answer:

• {3 - √2; 3 + √2}

Step-by-step explanation:

Given

• Quadratic function y = x² - 6x + 7

Solving by completing the square

• x² - 6x + 7 = 0                                    Given
• x² - 2*3x + 3² - 3² + 7 = 0                   Add, subtract 3²
• (x - 3)² - 9 + 7 = 0                                Simplify
• (x - 3)² - 2 = 0                                      Difference of squares
• (x - 3 + √2)(x - 3 - √2) = 0                   Factorize
• x - 3 + √2 = 0 and x - 3 - √2 = 0        Solve each root
• x = 3 - √2 and x = 3 + √2                   Answer