Solve for x
x=-\frac{1}{7}\approx -0.142857143
x=4
Graph
Share
Copied to clipboard
a+b=-27 ab=7\left(-4\right)=-28
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx-4. To find a and b, set up a system to be solved.
1,-28 2,-14 4,-7
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -28.
1-28=-27 2-14=-12 4-7=-3
Calculate the sum for each pair.
a=-28 b=1
The solution is the pair that gives sum -27.
\left(7x^{2}-28x\right)+\left(x-4\right)
Rewrite 7x^{2}-27x-4 as \left(7x^{2}-28x\right)+\left(x-4\right).
7x\left(x-4\right)+x-4
Factor out 7x in 7x^{2}-28x.
\left(x-4\right)\left(7x+1\right)
Factor out common term x-4 by using distributive property.
x=4 x=-\frac{1}{7}
To find equation solutions, solve x-4=0 and 7x+1=0.
7x^{2}-27x-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(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 7\left(-4\right)}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -27 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-27\right)±\sqrt{729-4\times 7\left(-4\right)}}{2\times 7}
Square -27.
x=\frac{-\left(-27\right)±\sqrt{729-28\left(-4\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-\left(-27\right)±\sqrt{729+112}}{2\times 7}
Multiply -28 times -4.
x=\frac{-\left(-27\right)±\sqrt{841}}{2\times 7}
Add 729 to 112.
x=\frac{-\left(-27\right)±29}{2\times 7}
Take the square root of 841.
x=\frac{27±29}{2\times 7}
The opposite of -27 is 27.
x=\frac{27±29}{14}
Multiply 2 times 7.
x=\frac{56}{14}
Now solve the equation x=\frac{27±29}{14} when ± is plus. Add 27 to 29.
x=4
Divide 56 by 14.
x=-\frac{2}{14}
Now solve the equation x=\frac{27±29}{14} when ± is minus. Subtract 29 from 27.
x=-\frac{1}{7}
Reduce the fraction \frac{-2}{14} to lowest terms by extracting and canceling out 2.
x=4 x=-\frac{1}{7}
The equation is now solved.
7x^{2}-27x-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.
7x^{2}-27x-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
7x^{2}-27x=-\left(-4\right)
Subtracting -4 from itself leaves 0.
7x^{2}-27x=4
Subtract -4 from 0.
\frac{7x^{2}-27x}{7}=\frac{4}{7}
Divide both sides by 7.
x^{2}-\frac{27}{7}x=\frac{4}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}-\frac{27}{7}x+\left(-\frac{27}{14}\right)^{2}=\frac{4}{7}+\left(-\frac{27}{14}\right)^{2}
Divide -\frac{27}{7}, the coefficient of the x term, by 2 to get -\frac{27}{14}. Then add the square of -\frac{27}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{27}{7}x+\frac{729}{196}=\frac{4}{7}+\frac{729}{196}
Square -\frac{27}{14} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{27}{7}x+\frac{729}{196}=\frac{841}{196}
Add \frac{4}{7} to \frac{729}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{27}{14}\right)^{2}=\frac{841}{196}
Factor x^{2}-\frac{27}{7}x+\frac{729}{196}. 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{27}{14}\right)^{2}}=\sqrt{\frac{841}{196}}
Take the square root of both sides of the equation.
x-\frac{27}{14}=\frac{29}{14} x-\frac{27}{14}=-\frac{29}{14}
Simplify.
x=4 x=-\frac{1}{7}
Add \frac{27}{14} to both sides of the equation.
x ^ 2 -\frac{27}{7}x -\frac{4}{7} = 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 7
r + s = \frac{27}{7} rs = -\frac{4}{7}
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{27}{14} - u s = \frac{27}{14} + u
Two numbers r and s sum up to \frac{27}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{27}{7} = \frac{27}{14}. 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{27}{14} - u) (\frac{27}{14} + u) = -\frac{4}{7}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{7}
\frac{729}{196} - u^2 = -\frac{4}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{7}-\frac{729}{196} = -\frac{841}{196}
Simplify the expression by subtracting \frac{729}{196} on both sides
u^2 = \frac{841}{196} u = \pm\sqrt{\frac{841}{196}} = \pm \frac{29}{14}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{27}{14} - \frac{29}{14} = -0.143 s = \frac{27}{14} + \frac{29}{14} = 4
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}