Solve 7left(1/2x^2+3right)=14x^2

$$7 \left( \frac{ 1 }{ 2 } { x }^{ 2 } +3 \right) = 14 { x }^{ 2 }$$

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

$$x=-\sqrt{2}\approx -1.414213562$$

Steps by Finding Square Root

Use the distributive property to multiply 7 by \frac{1}{2}x^{2}+3.

$$\frac{7}{2}x^{2}+21=14x^{2}$$

Subtract 14x^{2} from both sides.

$$\frac{7}{2}x^{2}+21-14x^{2}=0$$

Combine \frac{7}{2}x^{2} and -14x^{2} to get -\frac{21}{2}x^{2}.

$$-\frac{21}{2}x^{2}+21=0$$

Subtract 21 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{21}{2}x^{2}=-21$$

Multiply both sides by -\frac{2}{21}, the reciprocal of -\frac{21}{2}.

$$x^{2}=-21\left(-\frac{2}{21}\right)$$

Multiply -21 and -\frac{2}{21} to get 2.

$$x^{2}=2$$

Take the square root of both sides of the equation.

$$x=\sqrt{2} x=-\sqrt{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7 by \frac{1}{2}x^{2}+3.

$$\frac{7}{2}x^{2}+21=14x^{2}$$

Subtract 14x^{2} from both sides.

$$\frac{7}{2}x^{2}+21-14x^{2}=0$$

Combine \frac{7}{2}x^{2} and -14x^{2} to get -\frac{21}{2}x^{2}.

$$-\frac{21}{2}x^{2}+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{21}{2} for a, 0 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{21}{2}\right)\times 21}}{2\left(-\frac{21}{2}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{21}{2}\right)\times 21}}{2\left(-\frac{21}{2}\right)}$$

Multiply -4 times -\frac{21}{2}.

$$x=\frac{0±\sqrt{42\times 21}}{2\left(-\frac{21}{2}\right)}$$

Multiply 42 times 21.

$$x=\frac{0±\sqrt{882}}{2\left(-\frac{21}{2}\right)}$$

Take the square root of 882.

$$x=\frac{0±21\sqrt{2}}{2\left(-\frac{21}{2}\right)}$$

Multiply 2 times -\frac{21}{2}.

$$x=\frac{0±21\sqrt{2}}{-21}$$

Now solve the equation x=\frac{0±21\sqrt{2}}{-21} when ± is plus.

$$x=-\sqrt{2}$$

Now solve the equation x=\frac{0±21\sqrt{2}}{-21} when ± is minus.

$$x=\sqrt{2}$$

The equation is now solved.

$$x=-\sqrt{2} x=\sqrt{2}$$