Solve x^2=0.5+1.5x

$$x ^ { 2 } = 0.5 + 1.5 x$$

Solve for x

$$x = \frac{\sqrt{17} + 3}{4} \approx 1.780776406$$

$$x=\frac{3-\sqrt{17}}{4}\approx -0.280776406$$

Steps Using the Quadratic Formula

Subtract 0.5 from both sides.

$$x^{2}-0.5=1.5x$$

Subtract 1.5x from both sides.

$$x^{2}-0.5-1.5x=0$$

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.

$$x^{2}-1.5x-0.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1.5 for b, and -0.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.5\right)±\sqrt{\left(-1.5\right)^{2}-4\left(-0.5\right)}}{2}$$

Square -1.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.5\right)±\sqrt{2.25-4\left(-0.5\right)}}{2}$$

Multiply -4 times -0.5.

$$x=\frac{-\left(-1.5\right)±\sqrt{2.25+2}}{2}$$

Add 2.25 to 2.

$$x=\frac{-\left(-1.5\right)±\sqrt{4.25}}{2}$$

Take the square root of 4.25.

$$x=\frac{-\left(-1.5\right)±\frac{\sqrt{17}}{2}}{2}$$

The opposite of -1.5 is 1.5.

$$x=\frac{1.5±\frac{\sqrt{17}}{2}}{2}$$

Now solve the equation x=\frac{1.5±\frac{\sqrt{17}}{2}}{2} when ± is plus. Add 1.5 to \frac{\sqrt{17}}{2}.

$$x=\frac{\sqrt{17}+3}{2\times 2}$$

Divide \frac{3+\sqrt{17}}{2} by 2.

$$x=\frac{\sqrt{17}+3}{4}$$

Now solve the equation x=\frac{1.5±\frac{\sqrt{17}}{2}}{2} when ± is minus. Subtract \frac{\sqrt{17}}{2} from 1.5.

$$x=\frac{3-\sqrt{17}}{2\times 2}$$

Divide \frac{3-\sqrt{17}}{2} by 2.

$$x=\frac{3-\sqrt{17}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+3}{4} x=\frac{3-\sqrt{17}}{4}$$

Steps for Completing the Square

Subtract 1.5x from both sides.

$$x^{2}-1.5x=0.5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-1.5x=\frac{1}{2}$$

Divide -1.5, the coefficient of the x term, by 2 to get -0.75. Then add the square of -0.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.5x+\left(-0.75\right)^{2}=\frac{1}{2}+\left(-0.75\right)^{2}$$

Square -0.75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.5x+0.5625=\frac{1}{2}+0.5625$$

Add \frac{1}{2} to 0.5625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.5x+0.5625=\frac{17}{16}$$

Factor x^{2}-1.5x+0.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.75\right)^{2}=\frac{17}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.75\right)^{2}}=\sqrt{\frac{17}{16}}$$

Simplify.

$$x-0.75=\frac{\sqrt{17}}{4} x-0.75=-\frac{\sqrt{17}}{4}$$

Add 0.75 to both sides of the equation.

$$x=\frac{\sqrt{17}+3}{4} x=\frac{3-\sqrt{17}}{4}$$