Question

1. solve the equation 9x² + 12x + 4 = 0 using the quadratic equation.
2. solve the equation by factoring 2x² - 2x = 12

Explanation:

Step1: Identify coefficients for quadratic formula

For the equation $9x^{2}+12x + 4=0$, $a = 9$, $b = 12$, $c = 4$.

Step2: Calculate the discriminant $\Delta$

$\Delta=b^{2}-4ac=(12)^{2}-4\times9\times4=144 - 144=0$.

Step3: Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$

Since $\Delta = 0$, $x=\frac{-12\pm\sqrt{0}}{2\times9}=\frac{-12}{18}=-\frac{2}{3}$.

for second - part:

Step1: Rewrite the equation in standard form

Rewrite $2x^{2}-2x = 12$ as $2x^{2}-2x-12 = 0$. Divide through by 2 to get $x^{2}-x - 6=0$.

Step2: Factor the quadratic equation

We need two numbers that multiply to - 6 and add up to - 1. The numbers are - 3 and 2. So $x^{2}-x - 6=(x - 3)(x+2)=0$.

Step3: Solve for $x$

If $(x - 3)(x + 2)=0$, then $x-3=0$ or $x + 2=0$. So $x=3$ or $x=-2$.

Answer:

$x =-\frac{2}{3}$