Solve 28-(x+1)x=3

$$28-(x+1)x=3$$

Solve for x

$$x = \frac{\sqrt{101} - 1}{2} \approx 4.524937811$$

$$x=\frac{-\sqrt{101}-1}{2}\approx -5.524937811$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by x.

$$28-\left(x^{2}+x\right)=3$$

To find the opposite of x^{2}+x, find the opposite of each term.

$$28-x^{2}-x=3$$

Subtract 3 from both sides.

$$28-x^{2}-x-3=0$$

Subtract 3 from 28 to get 25.

$$25-x^{2}-x=0$$

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.

$$-x^{2}-x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 25}}{2\left(-1\right)}$$

Multiply 4 times 25.

$$x=\frac{-\left(-1\right)±\sqrt{1+100}}{2\left(-1\right)}$$

Add 1 to 100.

$$x=\frac{-\left(-1\right)±\sqrt{101}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{101}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{101}}{-2}$$

Now solve the equation x=\frac{1±\sqrt{101}}{-2} when ± is plus. Add 1 to \sqrt{101}.

$$x=\frac{\sqrt{101}+1}{-2}$$

Divide 1+\sqrt{101} by -2.

$$x=\frac{-\sqrt{101}-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{101}}{-2} when ± is minus. Subtract \sqrt{101} from 1.

$$x=\frac{1-\sqrt{101}}{-2}$$

Divide 1-\sqrt{101} by -2.

$$x=\frac{\sqrt{101}-1}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{101}-1}{2} x=\frac{\sqrt{101}-1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by x.

$$28-\left(x^{2}+x\right)=3$$

To find the opposite of x^{2}+x, find the opposite of each term.

$$28-x^{2}-x=3$$

Subtract 28 from both sides.

$$-x^{2}-x=3-28$$

Subtract 28 from 3 to get -25.

$$-x^{2}-x=-25$$

Divide both sides by -1.

$$\frac{-x^{2}-x}{-1}=-\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{1}{-1}\right)x=-\frac{25}{-1}$$

Divide -1 by -1.

$$x^{2}+x=-\frac{25}{-1}$$

Divide -25 by -1.

$$x^{2}+x=25$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=25+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=25+\frac{1}{4}$$

Add 25 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{101}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{101}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{101}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{101}}{2} x+\frac{1}{2}=-\frac{\sqrt{101}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{101}-1}{2} x=\frac{-\sqrt{101}-1}{2}$$