Skip to main content
Solve for a
Tick mark Image

Similar Problems from Web Search

Share

a^{2}-2a-\frac{15}{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.
a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-\frac{15}{2}\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -\frac{15}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-\left(-2\right)±\sqrt{4-4\left(-\frac{15}{2}\right)}}{2}
Square -2.
a=\frac{-\left(-2\right)±\sqrt{4+30}}{2}
Multiply -4 times -\frac{15}{2}.
a=\frac{-\left(-2\right)±\sqrt{34}}{2}
Add 4 to 30.
a=\frac{2±\sqrt{34}}{2}
The opposite of -2 is 2.
a=\frac{\sqrt{34}+2}{2}
Now solve the equation a=\frac{2±\sqrt{34}}{2} when ± is plus. Add 2 to \sqrt{34}.
a=\frac{\sqrt{34}}{2}+1
Divide 2+\sqrt{34} by 2.
a=\frac{2-\sqrt{34}}{2}
Now solve the equation a=\frac{2±\sqrt{34}}{2} when ± is minus. Subtract \sqrt{34} from 2.
a=-\frac{\sqrt{34}}{2}+1
Divide 2-\sqrt{34} by 2.
a=\frac{\sqrt{34}}{2}+1 a=-\frac{\sqrt{34}}{2}+1
The equation is now solved.
a^{2}-2a-\frac{15}{2}=0
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.
a^{2}-2a-\frac{15}{2}-\left(-\frac{15}{2}\right)=-\left(-\frac{15}{2}\right)
Add \frac{15}{2} to both sides of the equation.
a^{2}-2a=-\left(-\frac{15}{2}\right)
Subtracting -\frac{15}{2} from itself leaves 0.
a^{2}-2a=\frac{15}{2}
Subtract -\frac{15}{2} from 0.
a^{2}-2a+1=\frac{15}{2}+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}-2a+1=\frac{17}{2}
Add \frac{15}{2} to 1.
\left(a-1\right)^{2}=\frac{17}{2}
Factor a^{2}-2a+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(a-1\right)^{2}}=\sqrt{\frac{17}{2}}
Take the square root of both sides of the equation.
a-1=\frac{\sqrt{34}}{2} a-1=-\frac{\sqrt{34}}{2}
Simplify.
a=\frac{\sqrt{34}}{2}+1 a=-\frac{\sqrt{34}}{2}+1
Add 1 to both sides of the equation.