Solve 2x^2+19=14x

$$2 { x }^{ 2 } +19=14x$$

Solve for x

$$x = \frac{\sqrt{11} + 7}{2} \approx 5.158312395$$

$$x = \frac{7 - \sqrt{11}}{2} \approx 1.841687605$$

Steps Using the Quadratic Formula

Subtract 14x from both sides.

$$2x^{2}+19-14x=0$$

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.

$$2x^{2}-14x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -14 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 19}}{2\times 2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 19}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-14\right)±\sqrt{196-8\times 19}}{2\times 2}$$

Multiply -8 times 19.

$$x=\frac{-\left(-14\right)±\sqrt{196-152}}{2\times 2}$$

Add 196 to -152.

$$x=\frac{-\left(-14\right)±\sqrt{44}}{2\times 2}$$

Take the square root of 44.

$$x=\frac{-\left(-14\right)±2\sqrt{11}}{2\times 2}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{11}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{14±2\sqrt{11}}{4}$$

Now solve the equation x=\frac{14±2\sqrt{11}}{4} when ± is plus. Add 14 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+14}{4}$$

Divide 14+2\sqrt{11} by 4.

$$x=\frac{\sqrt{11}+7}{2}$$

Now solve the equation x=\frac{14±2\sqrt{11}}{4} when ± is minus. Subtract 2\sqrt{11} from 14.

$$x=\frac{14-2\sqrt{11}}{4}$$

Divide 14-2\sqrt{11} by 4.

$$x=\frac{7-\sqrt{11}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{11}+7}{2} x=\frac{7-\sqrt{11}}{2}$$

Steps for Completing the Square

Subtract 14x from both sides.

$$2x^{2}+19-14x=0$$

Subtract 19 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-14x=-19$$

Divide both sides by 2.

$$\frac{2x^{2}-14x}{2}=-\frac{19}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{14}{2}\right)x=-\frac{19}{2}$$

Divide -14 by 2.

$$x^{2}-7x=-\frac{19}{2}$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-\frac{19}{2}+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-\frac{19}{2}+\frac{49}{4}$$

Add -\frac{19}{2} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-7x+\frac{49}{4}=\frac{11}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{11}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{11}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{11}}{2} x-\frac{7}{2}=-\frac{\sqrt{11}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{11}+7}{2} x=\frac{7-\sqrt{11}}{2}$$