Solve 4x^2=15-14x

$$4 x ^ { 2 } = 15 - 14 x$$

Solve for x

$$x=\frac{\sqrt{109}-7}{4}\approx 0.860076627$$

$$x=\frac{-\sqrt{109}-7}{4}\approx -4.360076627$$

Steps Using the Quadratic Formula

Subtract 15 from both sides.

$$4x^{2}-15=-14x$$

Add 14x to both sides.

$$4x^{2}-15+14x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$4x^{2}+14x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 14 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 4\left(-15\right)}}{2\times 4}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 4\left(-15\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-14±\sqrt{196-16\left(-15\right)}}{2\times 4}$$

Multiply -16 times -15.

$$x=\frac{-14±\sqrt{196+240}}{2\times 4}$$

Add 196 to 240.

$$x=\frac{-14±\sqrt{436}}{2\times 4}$$

Take the square root of 436.

$$x=\frac{-14±2\sqrt{109}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-14±2\sqrt{109}}{8}$$

Now solve the equation x=\frac{-14±2\sqrt{109}}{8} when ± is plus. Add -14 to 2\sqrt{109}.

$$x=\frac{2\sqrt{109}-14}{8}$$

Divide -14+2\sqrt{109} by 8.

$$x=\frac{\sqrt{109}-7}{4}$$

Now solve the equation x=\frac{-14±2\sqrt{109}}{8} when ± is minus. Subtract 2\sqrt{109} from -14.

$$x=\frac{-2\sqrt{109}-14}{8}$$

Divide -14-2\sqrt{109} by 8.

$$x=\frac{-\sqrt{109}-7}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{109}-7}{4} x=\frac{-\sqrt{109}-7}{4}$$

Steps for Completing the Square

Add 14x to both sides.

$$4x^{2}+14x=15$$

Divide both sides by 4.

$$\frac{4x^{2}+14x}{4}=\frac{15}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{14}{4}x=\frac{15}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{2}x=\frac{15}{4}$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\frac{15}{4}+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{15}{4}+\frac{49}{16}$$

Add \frac{15}{4} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{109}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{109}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{109}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{\sqrt{109}}{4} x+\frac{7}{4}=-\frac{\sqrt{109}}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=\frac{\sqrt{109}-7}{4} x=\frac{-\sqrt{109}-7}{4}$$