Solve 1+a^2+a^2-55a+34=50

$$1 + a ^ { 2 } + a ^ { 2 } - 55 a + 34 = 50$$

Solve for a

$$a = \frac{\sqrt{3145} + 55}{4} \approx 27.770074893$$

$$a=\frac{55-\sqrt{3145}}{4}\approx -0.270074893$$

Steps Using the Quadratic Formula

Combine a^{2} and a^{2} to get 2a^{2}.

$$1+2a^{2}-55a+34=50$$

Add 1 and 34 to get 35.

$$35+2a^{2}-55a=50$$

Subtract 50 from both sides.

$$35+2a^{2}-55a-50=0$$

Subtract 50 from 35 to get -15.

$$-15+2a^{2}-55a=0$$

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.

$$2a^{2}-55a-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -55 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\times 2\left(-15\right)}}{2\times 2}$$

Square -55.

$$a=\frac{-\left(-55\right)±\sqrt{3025-4\times 2\left(-15\right)}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-55\right)±\sqrt{3025-8\left(-15\right)}}{2\times 2}$$

Multiply -8 times -15.

$$a=\frac{-\left(-55\right)±\sqrt{3025+120}}{2\times 2}$$

Add 3025 to 120.

$$a=\frac{-\left(-55\right)±\sqrt{3145}}{2\times 2}$$

The opposite of -55 is 55.

$$a=\frac{55±\sqrt{3145}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{55±\sqrt{3145}}{4}$$

Now solve the equation a=\frac{55±\sqrt{3145}}{4} when ± is plus. Add 55 to \sqrt{3145}.

$$a=\frac{\sqrt{3145}+55}{4}$$

Now solve the equation a=\frac{55±\sqrt{3145}}{4} when ± is minus. Subtract \sqrt{3145} from 55.

$$a=\frac{55-\sqrt{3145}}{4}$$

The equation is now solved.

$$a=\frac{\sqrt{3145}+55}{4} a=\frac{55-\sqrt{3145}}{4}$$

Steps for Completing the Square

Combine a^{2} and a^{2} to get 2a^{2}.

$$1+2a^{2}-55a+34=50$$

Add 1 and 34 to get 35.

$$35+2a^{2}-55a=50$$

Subtract 35 from both sides.

$$2a^{2}-55a=50-35$$

Subtract 35 from 50 to get 15.

$$2a^{2}-55a=15$$

Divide both sides by 2.

$$\frac{2a^{2}-55a}{2}=\frac{15}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}-\frac{55}{2}a=\frac{15}{2}$$

Divide -\frac{55}{2}, the coefficient of the x term, by 2 to get -\frac{55}{4}. Then add the square of -\frac{55}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{55}{2}a+\left(-\frac{55}{4}\right)^{2}=\frac{15}{2}+\left(-\frac{55}{4}\right)^{2}$$

Square -\frac{55}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{55}{2}a+\frac{3025}{16}=\frac{15}{2}+\frac{3025}{16}$$

Add \frac{15}{2} to \frac{3025}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{55}{2}a+\frac{3025}{16}=\frac{3145}{16}$$

Factor a^{2}-\frac{55}{2}a+\frac{3025}{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}.

$$\left(a-\frac{55}{4}\right)^{2}=\frac{3145}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{55}{4}\right)^{2}}=\sqrt{\frac{3145}{16}}$$

Simplify.

$$a-\frac{55}{4}=\frac{\sqrt{3145}}{4} a-\frac{55}{4}=-\frac{\sqrt{3145}}{4}$$

Add \frac{55}{4} to both sides of the equation.

$$a=\frac{\sqrt{3145}+55}{4} a=\frac{55-\sqrt{3145}}{4}$$