Algebra Examples

$x^{2} - \frac{4}{5} x = 3$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Combine $x$ and $\frac{4}{5}$ .

$x^{2} - \frac{x \cdot 4}{5} = 3$

Step 1.1.1.2

Move $4$ to the left of $x$ .

$x^{2} - \frac{4 x}{5} = 3$

Step 1.2

Subtract $3$ from both sides of the equation.

$x^{2} - \frac{4 x}{5} - 3 = 0$

Step 2

Multiply through by the least common denominator $5$ , then simplify.

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Step 2.1

Apply the distributive property.

$5 x^{2} + 5 (- \frac{4 x}{5}) + 5 \cdot - 3 = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{4 x}{5}$ into the numerator.

$5 x^{2} + 5 (\frac{- 4 x}{5}) + 5 \cdot - 3 = 0$

Step 2.2.1.2

Cancel the common factor.

$5 x^{2} + 5 (\frac{- 4 x}{5}) + 5 \cdot - 3 = 0$

Step 2.2.1.3

Rewrite the expression.

$5 x^{2} - 4 x + 5 \cdot - 3 = 0$

Step 2.2.2

Multiply $5$ by $- 3$ .

$5 x^{2} - 4 x - 15 = 0$

Step 3

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

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

Step 4

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

${(- 4)}^{2} - 4 (5 \cdot - 15)$

Step 5

Evaluate the result to find the discriminant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$16 - 4 (5 \cdot - 15)$

Step 5.1.2

Multiply $- 4 (5 \cdot - 15)$ .

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Step 5.1.2.1

Multiply $5$ by $- 15$ .

$16 - 4 \cdot - 75$

Step 5.1.2.2

Multiply $- 4$ by $- 75$ .

$16 + 300$

Step 5.2

Add $16$ and $300$ .

$316$

Step 6

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