Precalculus Examples

$7 x^{2} = 5 x + 3$

Step 1

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

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Step 1.1

Subtract $5 x$ from both sides of the equation.

$7 x^{2} - 5 x = 3$

Step 1.2

Subtract $3$ from both sides of the equation.

$7 x^{2} - 5 x - 3 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 7$ , $b = - 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (7 \cdot - 3)}}{2 \cdot 7}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 7 \cdot - 3}}{2 \cdot 7}$

Step 4.1.2

Multiply $- 4 \cdot 7 \cdot - 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{5 \pm \sqrt{25 - 28 \cdot - 3}}{2 \cdot 7}$

Step 4.1.2.2

Multiply $- 28$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 84}}{2 \cdot 7}$

Step 4.1.3

Add $25$ and $84$ .

$x = \frac{5 \pm \sqrt{109}}{2 \cdot 7}$

Step 4.2

Multiply $2$ by $7$ .

$x = \frac{5 \pm \sqrt{109}}{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{109}}{14}$

Decimal Form:

$x = 1.10287903 \dots, - 0.38859332 \dots$