Determine the Nature of the Roots Using the Discriminant 5x^2=7x+6

$5 x^{2} = 7 x + 6$

Step 1

Move all the expressions to the left side of the equation.

Subtract $7 x$ from both sides of the equation.

$5 x^{2} - 7 x = 6$

Subtract $6$ from both sides of the equation.

$5 x^{2} - 7 x - 6 = 0$

Step 2

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 3

Substitute in the values of $a$ , $b$ , and $c$ .

${(- 7)}^{2} - 4 (5 \cdot - 6)$

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$49 - 4 (5 \cdot - 6)$

Multiply $- 4 (5 \cdot - 6)$ .

Multiply $5$ by $- 6$ .

$49 - 4 \cdot - 30$

Multiply $- 4$ by $- 30$ .

$49 + 120$

Add $49$ and $120$ .

$169$

Step 5

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant $(Δ)$ :

$Δ > 0$ means there are $2$ distinct real roots.

$Δ = 0$ means there are $2$ equal real roots, or $1$ distinct real root.

$Δ < 0$ means there are no real roots, but $2$ complex roots.

Since the discriminant is greater than $0$ , there are two real roots.

Two Real Roots