Factor
\left(x-\frac{-\sqrt{65}-7}{2}\right)\left(x-\frac{\sqrt{65}-7}{2}\right)
Evaluate
x^{2}+7x-4
Graph
Share
Copied to clipboard
x^{2}+7x-4=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-7±\sqrt{7^{2}-4\left(-4\right)}}{2}
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{-7±\sqrt{49-4\left(-4\right)}}{2}
Square 7.
x=\frac{-7±\sqrt{49+16}}{2}
Multiply -4 times -4.
x=\frac{-7±\sqrt{65}}{2}
Add 49 to 16.
x=\frac{\sqrt{65}-7}{2}
Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is plus. Add -7 to \sqrt{65}.
x=\frac{-\sqrt{65}-7}{2}
Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is minus. Subtract \sqrt{65} from -7.
x^{2}+7x-4=\left(x-\frac{\sqrt{65}-7}{2}\right)\left(x-\frac{-\sqrt{65}-7}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{65}}{2} for x_{1} and \frac{-7-\sqrt{65}}{2} for x_{2}.
x ^ 2 +7x -4 = 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.
r + s = -7 rs = -4
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{7}{2} - u s = -\frac{7}{2} + u
Two numbers r and s sum up to -7 exactly when the average of the two numbers is \frac{1}{2}*-7 = -\frac{7}{2}. 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{7}{2} - u) (-\frac{7}{2} + u) = -4
To solve for unknown quantity u, substitute these in the product equation rs = -4
\frac{49}{4} - u^2 = -4
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -4-\frac{49}{4} = -\frac{65}{4}
Simplify the expression by subtracting \frac{49}{4} on both sides
u^2 = \frac{65}{4} u = \pm\sqrt{\frac{65}{4}} = \pm \frac{\sqrt{65}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{7}{2} - \frac{\sqrt{65}}{2} = -7.531 s = -\frac{7}{2} + \frac{\sqrt{65}}{2} = 0.531
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}