Question

Use the discriminant to describe the solutions as one real, two real, or two imaginary solutions.

x^2 - 15x + 12 = 0

Answer 1

Answer: Two real solutions

1. Identify the coefficients , and of the quadratic equation.

a=1, b= -15, c=12

2. Evaluate the discriminant by substituting , and into the expression.

(-15)^2 - 4 • 1 • 12

3. Simplify the expression

177 - The discriminant is greater than 0, so the quadratic equation has two real solutions.

Answer 2

2 real solutions.

Explanation:

Discriminant

`\boxed{b^2-4ac}\quad\textsf{when}\:ax^2+bx+c=0`

`\textsf{when }\:b^2-4ac > 0 \implies \textsf{two real solutions}.`

`\textsf{when }\:b^2-4ac=0 \implies \textsf{one real solution}.`

`\textsf{when }\:b^2-4ac < 0 \implies \textsf{no real solutions}.`

Given equation:

`x^2-15x+12=0`

Therefore:

• a = 1
• b = -15
• c = 12

Substitute these values into the discriminant formula:

`\begin{aligned}\implies b^2-4ac&=(-15)^2-4(1)(12)\\&=225-48\\&=177\end{aligned}`

Therefore, as b² - 4ac > 0 there are two real solutions.