Solve 3x^2-149*2x+400=0

$$3 x ^ { 2 } - 149 \times 2 x + 400 = 0$$

Solve for x

$$x = \frac{\sqrt{21001} + 149}{3} \approx 97.972405916$$

$$x = \frac{149 - \sqrt{21001}}{3} \approx 1.360927417$$

Steps Using the Quadratic Formula

Multiply 149 and 2 to get 298.

$$3x^{2}-298x+400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -298 for b, and 400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-298\right)±\sqrt{\left(-298\right)^{2}-4\times 3\times 400}}{2\times 3}$$

Square -298.

$$x=\frac{-\left(-298\right)±\sqrt{88804-4\times 3\times 400}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-298\right)±\sqrt{88804-12\times 400}}{2\times 3}$$

Multiply -12 times 400.

$$x=\frac{-\left(-298\right)±\sqrt{88804-4800}}{2\times 3}$$

Add 88804 to -4800.

$$x=\frac{-\left(-298\right)±\sqrt{84004}}{2\times 3}$$

Take the square root of 84004.

$$x=\frac{-\left(-298\right)±2\sqrt{21001}}{2\times 3}$$

The opposite of -298 is 298.

$$x=\frac{298±2\sqrt{21001}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{298±2\sqrt{21001}}{6}$$

Now solve the equation x=\frac{298±2\sqrt{21001}}{6} when ± is plus. Add 298 to 2\sqrt{21001}.

$$x=\frac{2\sqrt{21001}+298}{6}$$

Divide 298+2\sqrt{21001} by 6.

$$x=\frac{\sqrt{21001}+149}{3}$$

Now solve the equation x=\frac{298±2\sqrt{21001}}{6} when ± is minus. Subtract 2\sqrt{21001} from 298.

$$x=\frac{298-2\sqrt{21001}}{6}$$

Divide 298-2\sqrt{21001} by 6.

$$x=\frac{149-\sqrt{21001}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{21001}+149}{3} x=\frac{149-\sqrt{21001}}{3}$$

Steps for Completing the Square

Multiply 149 and 2 to get 298.

$$3x^{2}-298x+400=0$$

Subtract 400 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-298x=-400$$

Divide both sides by 3.

$$\frac{3x^{2}-298x}{3}=-\frac{400}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{298}{3}x=-\frac{400}{3}$$

Divide -\frac{298}{3}, the coefficient of the x term, by 2 to get -\frac{149}{3}. Then add the square of -\frac{149}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{298}{3}x+\left(-\frac{149}{3}\right)^{2}=-\frac{400}{3}+\left(-\frac{149}{3}\right)^{2}$$

Square -\frac{149}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{298}{3}x+\frac{22201}{9}=-\frac{400}{3}+\frac{22201}{9}$$

Add -\frac{400}{3} to \frac{22201}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{298}{3}x+\frac{22201}{9}=\frac{21001}{9}$$

Factor x^{2}-\frac{298}{3}x+\frac{22201}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{149}{3}\right)^{2}=\frac{21001}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{149}{3}\right)^{2}}=\sqrt{\frac{21001}{9}}$$

Simplify.

$$x-\frac{149}{3}=\frac{\sqrt{21001}}{3} x-\frac{149}{3}=-\frac{\sqrt{21001}}{3}$$

Add \frac{149}{3} to both sides of the equation.

$$x=\frac{\sqrt{21001}+149}{3} x=\frac{149-\sqrt{21001}}{3}$$