Solve 7*5x^2+3x=14

$$7 \times 5 { x }^{ 2 } +3x=14$$

Solve for x

$$x=\frac{\sqrt{1969}-3}{70}\approx 0.591048792$$

$$x=\frac{-\sqrt{1969}-3}{70}\approx -0.676763078$$

Steps Using the Quadratic Formula

Multiply 7 and 5 to get 35.

$$35x^{2}+3x=14$$

Subtract 14 from both sides.

$$35x^{2}+3x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, 3 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 35\left(-14\right)}}{2\times 35}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 35\left(-14\right)}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-3±\sqrt{9-140\left(-14\right)}}{2\times 35}$$

Multiply -140 times -14.

$$x=\frac{-3±\sqrt{9+1960}}{2\times 35}$$

Add 9 to 1960.

$$x=\frac{-3±\sqrt{1969}}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{-3±\sqrt{1969}}{70}$$

Now solve the equation x=\frac{-3±\sqrt{1969}}{70} when ± is plus. Add -3 to \sqrt{1969}.

$$x=\frac{\sqrt{1969}-3}{70}$$

Now solve the equation x=\frac{-3±\sqrt{1969}}{70} when ± is minus. Subtract \sqrt{1969} from -3.

$$x=\frac{-\sqrt{1969}-3}{70}$$

The equation is now solved.

$$x=\frac{\sqrt{1969}-3}{70} x=\frac{-\sqrt{1969}-3}{70}$$

Steps for Completing the Square

Multiply 7 and 5 to get 35.

$$35x^{2}+3x=14$$

Divide both sides by 35.

$$\frac{35x^{2}+3x}{35}=\frac{14}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}+\frac{3}{35}x=\frac{14}{35}$$

Reduce the fraction \frac{14}{35} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{3}{35}x=\frac{2}{5}$$

Divide \frac{3}{35}, the coefficient of the x term, by 2 to get \frac{3}{70}. Then add the square of \frac{3}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{35}x+\left(\frac{3}{70}\right)^{2}=\frac{2}{5}+\left(\frac{3}{70}\right)^{2}$$

Square \frac{3}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{35}x+\frac{9}{4900}=\frac{2}{5}+\frac{9}{4900}$$

Add \frac{2}{5} to \frac{9}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{35}x+\frac{9}{4900}=\frac{1969}{4900}$$

Factor x^{2}+\frac{3}{35}x+\frac{9}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{70}\right)^{2}=\frac{1969}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{70}\right)^{2}}=\sqrt{\frac{1969}{4900}}$$

Simplify.

$$x+\frac{3}{70}=\frac{\sqrt{1969}}{70} x+\frac{3}{70}=-\frac{\sqrt{1969}}{70}$$

Subtract \frac{3}{70} from both sides of the equation.

$$x=\frac{\sqrt{1969}-3}{70} x=\frac{-\sqrt{1969}-3}{70}$$