Solve 10x^2-65x+075=0

$$10 x ^ { 2 } - 65 x + 075 = 0$$

Solve for x

$$x = \frac{13}{2} = 6\frac{1}{2} = 6.5$$

$$x=0$$

Steps Using Factoring

Multiply 0 and 75 to get 0.

$$10x^{2}-65x+0=0$$

Anything plus zero gives itself.

$$10x^{2}-65x=0$$

Factor out x.

$$x\left(10x-65\right)=0$$

To find equation solutions, solve x=0 and 10x-65=0.

$$x=0 x=\frac{13}{2}$$

Steps Using the Quadratic Formula

Multiply 0 and 75 to get 0.

$$10x^{2}-65x+0=0$$

Anything plus zero gives itself.

$$10x^{2}-65x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -65 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-65\right)±\sqrt{\left(-65\right)^{2}}}{2\times 10}$$

Take the square root of \left(-65\right)^{2}.

$$x=\frac{-\left(-65\right)±65}{2\times 10}$$

The opposite of -65 is 65.

$$x=\frac{65±65}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{65±65}{20}$$

Now solve the equation x=\frac{65±65}{20} when ± is plus. Add 65 to 65.

$$x=\frac{130}{20}$$

Reduce the fraction \frac{130}{20} to lowest terms by extracting and canceling out 10.

$$x=\frac{13}{2}$$

Now solve the equation x=\frac{65±65}{20} when ± is minus. Subtract 65 from 65.

$$x=\frac{0}{20}$$

Divide 0 by 20.

$$x=0$$

The equation is now solved.

$$x=\frac{13}{2} x=0$$

Steps for Completing the Square

Multiply 0 and 75 to get 0.

$$10x^{2}-65x+0=0$$

Anything plus zero gives itself.

$$10x^{2}-65x=0$$

Divide both sides by 10.

$$\frac{10x^{2}-65x}{10}=\frac{0}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{65}{10}\right)x=\frac{0}{10}$$

Reduce the fraction \frac{-65}{10} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{13}{2}x=\frac{0}{10}$$

Divide 0 by 10.

$$x^{2}-\frac{13}{2}x=0$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{169}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{13}{4} x-\frac{13}{4}=-\frac{13}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$x=\frac{13}{2} x=0$$