Question

1. what are the solutions to the equation:

$y = 5x^2 + x - 6$?
a. $x = 1$ is the only solution
b. $x = -1$ and $x = \frac{6}{5}$
c. $x = 1$ and $x = -\frac{6}{5}$
d. $x = -1$ and $x = -\frac{5}{6}$

Explanation:

Step1: Set \( y = 0 \)

To find the solutions of the equation \( y = 5x^2 + x - 6 \), we set \( y = 0 \), so we have the quadratic equation \( 5x^2 + x - 6 = 0 \).

Step2: Factor the quadratic

We factor the quadratic expression. We need two numbers that multiply to \( 5\times(-6)= -30 \) and add up to \( 1 \). The numbers are \( 6 \) and \( -5 \). So we rewrite the middle term:

$$
LATEXBLOCK0
$$

Step3: Solve for \( x \)

Set each factor equal to zero:

• For \( 5x + 6 = 0 \), we get \( 5x = -6 \), so \( x = -\frac{6}{5} \).
• For \( x - 1 = 0 \), we get \( x = 1 \).

Answer:

c. \( x = 1 \) and \( x = -\frac{6}{5} \)