Solve for x
x = \frac{\sqrt{145} + 9}{8} \approx 2.630199322
x=\frac{9-\sqrt{145}}{8}\approx -0.380199322
Graph
Share
Copied to clipboard
4x^{2}-9x-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.
x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\left(-4\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -9 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\left(-4\right)}}{2\times 4}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81-16\left(-4\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-9\right)±\sqrt{81+64}}{2\times 4}
Multiply -16 times -4.
x=\frac{-\left(-9\right)±\sqrt{145}}{2\times 4}
Add 81 to 64.
x=\frac{9±\sqrt{145}}{2\times 4}
The opposite of -9 is 9.
x=\frac{9±\sqrt{145}}{8}
Multiply 2 times 4.
x=\frac{\sqrt{145}+9}{8}
Now solve the equation x=\frac{9±\sqrt{145}}{8} when ± is plus. Add 9 to \sqrt{145}.
x=\frac{9-\sqrt{145}}{8}
Now solve the equation x=\frac{9±\sqrt{145}}{8} when ± is minus. Subtract \sqrt{145} from 9.
x=\frac{\sqrt{145}+9}{8} x=\frac{9-\sqrt{145}}{8}
The equation is now solved.
4x^{2}-9x-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.
4x^{2}-9x-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
4x^{2}-9x=-\left(-4\right)
Subtracting -4 from itself leaves 0.
4x^{2}-9x=4
Subtract -4 from 0.
\frac{4x^{2}-9x}{4}=\frac{4}{4}
Divide both sides by 4.
x^{2}-\frac{9}{4}x=\frac{4}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{9}{4}x=1
Divide 4 by 4.
x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=1+\left(-\frac{9}{8}\right)^{2}
Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{9}{4}x+\frac{81}{64}=1+\frac{81}{64}
Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{145}{64}
Add 1 to \frac{81}{64}.
\left(x-\frac{9}{8}\right)^{2}=\frac{145}{64}
Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. 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(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{145}{64}}
Take the square root of both sides of the equation.
x-\frac{9}{8}=\frac{\sqrt{145}}{8} x-\frac{9}{8}=-\frac{\sqrt{145}}{8}
Simplify.
x=\frac{\sqrt{145}+9}{8} x=\frac{9-\sqrt{145}}{8}
Add \frac{9}{8} to both sides of the equation.
x ^ 2 -\frac{9}{4}x -1 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4
r + s = \frac{9}{4} rs = -1
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = \frac{9}{8} - u s = \frac{9}{8} + u
Two numbers r and s sum up to \frac{9}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{4} = \frac{9}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(\frac{9}{8} - u) (\frac{9}{8} + u) = -1
To solve for unknown quantity u, substitute these in the product equation rs = -1
\frac{81}{64} - u^2 = -1
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -1-\frac{81}{64} = -\frac{145}{64}
Simplify the expression by subtracting \frac{81}{64} on both sides
u^2 = \frac{145}{64} u = \pm\sqrt{\frac{145}{64}} = \pm \frac{\sqrt{145}}{8}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{9}{8} - \frac{\sqrt{145}}{8} = -0.380 s = \frac{9}{8} + \frac{\sqrt{145}}{8} = 2.630
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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}