Question

1. by substituting p = 10x + 7, solve for x:

(10x + 7)^2 + 19(10x + 7) + 90 = 0
state your solutions for x on the same line, separated by a comma.
enter your next step here

Explanation:

Step1: Substitute \( p = 10x + 7 \)

The equation becomes \( p^2 + 19p + 90 = 0 \)

Step2: Factor the quadratic equation

We need two numbers that multiply to \( 90 \) and add to \( 19 \). The numbers are \( 9 \) and \( 10 \). So, \( p^2 + 19p + 90=(p + 9)(p + 10)=0 \)

Step3: Solve for \( p \)

Set each factor equal to zero: \( p + 9 = 0 \) or \( p + 10 = 0 \), so \( p=-9 \) or \( p = - 10 \)

Step4: Substitute back \( p = 10x + 7 \) and solve for \( x \)

• For \( p=-9 \): \( 10x + 7=-9 \)

Subtract \( 7 \) from both sides: \( 10x=-9 - 7=-16 \)
Divide by \( 10 \): \( x=\frac{-16}{10}=-\frac{8}{5}=-1.6 \)

• For \( p = - 10 \): \( 10x + 7=-10 \)

Subtract \( 7 \) from both sides: \( 10x=-10 - 7=-17 \)
Divide by \( 10 \): \( x=\frac{-17}{10}=-1.7 \)

Answer:

\(-1.7, -1.6\)