Algebra Examples

$9 x^{2} - 6 x = - 9$

Step 1

Add $9$ to both sides of the equation.

$9 x^{2} - 6 x + 9 = 0$

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 $c$ .

${(- 6)}^{2} - 4 (9 \cdot 9)$

Step 4

Evaluate the result to find the discriminant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$36 - 4 (9 \cdot 9)$

Step 4.1.2

Multiply $- 4 (9 \cdot 9)$ .

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Step 4.1.2.1

Multiply $9$ by $9$ .

$36 - 4 \cdot 81$

Step 4.1.2.2

Multiply $- 4$ by $81$ .

$36 - 324$

Step 4.2

Subtract $324$ from $36$ .

$- 288$

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 less than $0$ there are no real roots. Instead, there are two complex roots.

Two Complex Roots