Solve for x
x=-1
x=4
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x^{2}-3x-4=0
Divide both sides by 6.
a+b=-3 ab=1\left(-4\right)=-4
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-4. To find a and b, set up a system to be solved.
1,-4 2,-2
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 -4.
1-4=-3 2-2=0
Calculate the sum for each pair.
a=-4 b=1
The solution is the pair that gives sum -3.
\left(x^{2}-4x\right)+\left(x-4\right)
Rewrite x^{2}-3x-4 as \left(x^{2}-4x\right)+\left(x-4\right).
x\left(x-4\right)+x-4
Factor out x in x^{2}-4x.
\left(x-4\right)\left(x+1\right)
Factor out common term x-4 by using distributive property.
x=4 x=-1
To find equation solutions, solve x-4=0 and x+1=0.
6x^{2}-18x-24=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(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 6\left(-24\right)}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -18 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 6\left(-24\right)}}{2\times 6}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-24\left(-24\right)}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-18\right)±\sqrt{324+576}}{2\times 6}
Multiply -24 times -24.
x=\frac{-\left(-18\right)±\sqrt{900}}{2\times 6}
Add 324 to 576.
x=\frac{-\left(-18\right)±30}{2\times 6}
Take the square root of 900.
x=\frac{18±30}{2\times 6}
The opposite of -18 is 18.
x=\frac{18±30}{12}
Multiply 2 times 6.
x=\frac{48}{12}
Now solve the equation x=\frac{18±30}{12} when ± is plus. Add 18 to 30.
x=4
Divide 48 by 12.
x=-\frac{12}{12}
Now solve the equation x=\frac{18±30}{12} when ± is minus. Subtract 30 from 18.
x=-1
Divide -12 by 12.
x=4 x=-1
The equation is now solved.
6x^{2}-18x-24=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.
6x^{2}-18x-24-\left(-24\right)=-\left(-24\right)
Add 24 to both sides of the equation.
6x^{2}-18x=-\left(-24\right)
Subtracting -24 from itself leaves 0.
6x^{2}-18x=24
Subtract -24 from 0.
\frac{6x^{2}-18x}{6}=\frac{24}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{18}{6}\right)x=\frac{24}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-3x=\frac{24}{6}
Divide -18 by 6.
x^{2}-3x=4
Divide 24 by 6.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=4+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=\frac{25}{4}
Add 4 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}
Factor x^{2}-3x+\frac{9}{4}. 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{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{5}{2} x-\frac{3}{2}=-\frac{5}{2}
Simplify.
x=4 x=-1
Add \frac{3}{2} to both sides of the equation.
x ^ 2 -3x -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.This is achieved by dividing both sides of the equation by 6
r + s = 3 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{3}{2} - u s = \frac{3}{2} + u
Two numbers r and s sum up to 3 exactly when the average of the two numbers is \frac{1}{2}*3 = \frac{3}{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{3}{2} - u) (\frac{3}{2} + u) = -4
To solve for unknown quantity u, substitute these in the product equation rs = -4
\frac{9}{4} - u^2 = -4
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -4-\frac{9}{4} = -\frac{25}{4}
Simplify the expression by subtracting \frac{9}{4} on both sides
u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{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{3}{2} - \frac{5}{2} = -1 s = \frac{3}{2} + \frac{5}{2} = 4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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Simultaneous equation
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Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
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Limits
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