Solve for a
a=\frac{-1+\sqrt{15}i}{4}\approx -0.25+0.968245837i
a=\frac{-\sqrt{15}i-1}{4}\approx -0.25-0.968245837i
Share
Copied to clipboard
4a^{2}+2a+4=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{-2±\sqrt{2^{2}-4\times 4\times 4}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 2 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-2±\sqrt{4-4\times 4\times 4}}{2\times 4}
Square 2.
a=\frac{-2±\sqrt{4-16\times 4}}{2\times 4}
Multiply -4 times 4.
a=\frac{-2±\sqrt{4-64}}{2\times 4}
Multiply -16 times 4.
a=\frac{-2±\sqrt{-60}}{2\times 4}
Add 4 to -64.
a=\frac{-2±2\sqrt{15}i}{2\times 4}
Take the square root of -60.
a=\frac{-2±2\sqrt{15}i}{8}
Multiply 2 times 4.
a=\frac{-2+2\sqrt{15}i}{8}
Now solve the equation a=\frac{-2±2\sqrt{15}i}{8} when ± is plus. Add -2 to 2i\sqrt{15}.
a=\frac{-1+\sqrt{15}i}{4}
Divide -2+2i\sqrt{15} by 8.
a=\frac{-2\sqrt{15}i-2}{8}
Now solve the equation a=\frac{-2±2\sqrt{15}i}{8} when ± is minus. Subtract 2i\sqrt{15} from -2.
a=\frac{-\sqrt{15}i-1}{4}
Divide -2-2i\sqrt{15} by 8.
a=\frac{-1+\sqrt{15}i}{4} a=\frac{-\sqrt{15}i-1}{4}
The equation is now solved.
4a^{2}+2a+4=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.
4a^{2}+2a+4-4=-4
Subtract 4 from both sides of the equation.
4a^{2}+2a=-4
Subtracting 4 from itself leaves 0.
\frac{4a^{2}+2a}{4}=-\frac{4}{4}
Divide both sides by 4.
a^{2}+\frac{2}{4}a=-\frac{4}{4}
Dividing by 4 undoes the multiplication by 4.
a^{2}+\frac{1}{2}a=-\frac{4}{4}
Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.
a^{2}+\frac{1}{2}a=-1
Divide -4 by 4.
a^{2}+\frac{1}{2}a+\left(\frac{1}{4}\right)^{2}=-1+\left(\frac{1}{4}\right)^{2}
Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}+\frac{1}{2}a+\frac{1}{16}=-1+\frac{1}{16}
Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.
a^{2}+\frac{1}{2}a+\frac{1}{16}=-\frac{15}{16}
Add -1 to \frac{1}{16}.
\left(a+\frac{1}{4}\right)^{2}=-\frac{15}{16}
Factor a^{2}+\frac{1}{2}a+\frac{1}{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}.
\sqrt{\left(a+\frac{1}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}
Take the square root of both sides of the equation.
a+\frac{1}{4}=\frac{\sqrt{15}i}{4} a+\frac{1}{4}=-\frac{\sqrt{15}i}{4}
Simplify.
a=\frac{-1+\sqrt{15}i}{4} a=\frac{-\sqrt{15}i-1}{4}
Subtract \frac{1}{4} from both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}