Solve x^2-54x+97=7

$${ x }^{ 2 } -54x+97=7$$

Solve for x

$$x=3\sqrt{71}+27\approx 52.27844932$$

$$x=27-3\sqrt{71}\approx 1.72155068$$

Steps Using the Quadratic Formula

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.

$$x^{2}-54x+97=7$$

Subtract 7 from both sides of the equation.

$$x^{2}-54x+97-7=7-7$$

Subtracting 7 from itself leaves 0.

$$x^{2}-54x+97-7=0$$

Subtract 7 from 97.

$$x^{2}-54x+90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -54 for b, and 90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 90}}{2}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 90}}{2}$$

Multiply -4 times 90.

$$x=\frac{-\left(-54\right)±\sqrt{2916-360}}{2}$$

Add 2916 to -360.

$$x=\frac{-\left(-54\right)±\sqrt{2556}}{2}$$

Take the square root of 2556.

$$x=\frac{-\left(-54\right)±6\sqrt{71}}{2}$$

The opposite of -54 is 54.

$$x=\frac{54±6\sqrt{71}}{2}$$

Now solve the equation x=\frac{54±6\sqrt{71}}{2} when ± is plus. Add 54 to 6\sqrt{71}.

$$x=\frac{6\sqrt{71}+54}{2}$$

Divide 54+6\sqrt{71} by 2.

$$x=3\sqrt{71}+27$$

Now solve the equation x=\frac{54±6\sqrt{71}}{2} when ± is minus. Subtract 6\sqrt{71} from 54.

$$x=\frac{54-6\sqrt{71}}{2}$$

Divide 54-6\sqrt{71} by 2.

$$x=27-3\sqrt{71}$$

The equation is now solved.

$$x=3\sqrt{71}+27 x=27-3\sqrt{71}$$

Steps for Completing the Square

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.

$$x^{2}-54x+97=7$$

Subtract 97 from both sides of the equation.

$$x^{2}-54x+97-97=7-97$$

Subtracting 97 from itself leaves 0.

$$x^{2}-54x=7-97$$

Subtract 97 from 7.

$$x^{2}-54x=-90$$

Divide -54, the coefficient of the x term, by 2 to get -27. Then add the square of -27 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-54x+\left(-27\right)^{2}=-90+\left(-27\right)^{2}$$

Square -27.

$$x^{2}-54x+729=-90+729$$

Add -90 to 729.

$$x^{2}-54x+729=639$$

Factor x^{2}-54x+729. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-27\right)^{2}=639$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-27\right)^{2}}=\sqrt{639}$$

Simplify.

$$x-27=3\sqrt{71} x-27=-3\sqrt{71}$$

Add 27 to both sides of the equation.

$$x=3\sqrt{71}+27 x=27-3\sqrt{71}$$