Solve x-1=x^2-49

$$x-1= { x }^{ 2 } -49$$

Solve for x

$$x = \frac{\sqrt{193} + 1}{2} \approx 7.446221995$$

$$x=\frac{1-\sqrt{193}}{2}\approx -6.446221995$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-1-x^{2}=-49$$

Add 49 to both sides.

$$x-1-x^{2}+49=0$$

Add -1 and 49 to get 48.

$$x+48-x^{2}=0$$

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.

$$-x^{2}+x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1±\sqrt{1+4\times 48}}{2\left(-1\right)}$$

Multiply 4 times 48.

$$x=\frac{-1±\sqrt{1+192}}{2\left(-1\right)}$$

Add 1 to 192.

$$x=\frac{-1±\sqrt{193}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\sqrt{193}}{-2}$$

Now solve the equation x=\frac{-1±\sqrt{193}}{-2} when ± is plus. Add -1 to \sqrt{193}.

$$x=\frac{\sqrt{193}-1}{-2}$$

Divide -1+\sqrt{193} by -2.

$$x=\frac{1-\sqrt{193}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{193}}{-2} when ± is minus. Subtract \sqrt{193} from -1.

$$x=\frac{-\sqrt{193}-1}{-2}$$

Divide -1-\sqrt{193} by -2.

$$x=\frac{\sqrt{193}+1}{2}$$

The equation is now solved.

$$x=\frac{1-\sqrt{193}}{2} x=\frac{\sqrt{193}+1}{2}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-1-x^{2}=-49$$

Add 1 to both sides.

$$x-x^{2}=-49+1$$

Add -49 and 1 to get -48.

$$x-x^{2}=-48$$

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.

$$-x^{2}+x=-48$$

Divide both sides by -1.

$$\frac{-x^{2}+x}{-1}=-\frac{48}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1}{-1}x=-\frac{48}{-1}$$

Divide 1 by -1.

$$x^{2}-x=-\frac{48}{-1}$$

Divide -48 by -1.

$$x^{2}-x=48$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=48+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=48+\frac{1}{4}$$

Add 48 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{193}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{193}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{193}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{193}}{2} x-\frac{1}{2}=-\frac{\sqrt{193}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{193}+1}{2} x=\frac{1-\sqrt{193}}{2}$$