Solve by factoring. 22. x^2-9x=0 23. 2x+24=-14x Show work please

Question

asked May 16, 2024 681 views

1 Answer

Matthew King · Answer 1 · 2024-05-20T08:45:44+0000

Answer:

$\textsf{22.}\quad x = 0, \;\;x = 9$

$\textsf{23.}\quad x = -3, \;\;x = -4$

Explanation:

To solve the quadratic equation x² - 9x = 0 by factoring, start by factoring out the common factor, which is x:

x(x - 9) = 0

Now, set each factor equal to zero and solve for x:

$\textsf{Factor 1:} \qquad\;\;\;x = 0$

$\begin{aligned}\textsf{Factor 2:} \qquad\;\;\;x - 9 &= 0\\x - 9 + 9 &= 0 + 9\\x &= 9\end{aligned}$

Therefore, the solutions to the equation are:

$\large\boxed{\boxed{x = 0\;\textsf{and}\;x = 9}}$

$\hrulefill$

To solve the quadratic equation 2x² + 24 = -14x by factoring, begin by adding 14x to both sides of the equation:

$\begin{aligned}2x^2+24+14x&=-14x+14x\\2x^2+14x+24&=0\end{aligned}$

Divide the entire equation by 2 to simplify it:

x^2+7x+12=0

To factor a quadratic in the form ax² + bx + c, we need to find two numbers that multiply to ac and sum to b.

Two numbers that multiply to 12 and sum to 7 are 3 and 4.

Rewrite the middle term as the sum of these two numbers:

x^2+3x+4x+12=0

Factor the first two terms and the last two terms separately:

x(x+3)+4(x+3)=0

Factor out the common term (x + 3):

(x+3)(x+4)=0

Apply the zero-product property:

$x+3=0 \implies x=-3$

$x+4=0 \implies x=-4$

Therefore, the solutions to the equation are:

$\large\boxed{\boxed{x = -3\;\textsf{and}\;x = -4}}$