Skip to main content
Solve for x
Tick mark Image
Graph

Similar Problems from Web Search

Share

a+b=-6 ab=-7
To solve the equation, factor x^{2}-6x-7 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
a=-7 b=1
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. The only such pair is the system solution.
\left(x-7\right)\left(x+1\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=7 x=-1
To find equation solutions, solve x-7=0 and x+1=0.
a+b=-6 ab=1\left(-7\right)=-7
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-7. To find a and b, set up a system to be solved.
a=-7 b=1
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. The only such pair is the system solution.
\left(x^{2}-7x\right)+\left(x-7\right)
Rewrite x^{2}-6x-7 as \left(x^{2}-7x\right)+\left(x-7\right).
x\left(x-7\right)+x-7
Factor out x in x^{2}-7x.
\left(x-7\right)\left(x+1\right)
Factor out common term x-7 by using distributive property.
x=7 x=-1
To find equation solutions, solve x-7=0 and x+1=0.
x^{2}-6x-7=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(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-7\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\left(-7\right)}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36+28}}{2}
Multiply -4 times -7.
x=\frac{-\left(-6\right)±\sqrt{64}}{2}
Add 36 to 28.
x=\frac{-\left(-6\right)±8}{2}
Take the square root of 64.
x=\frac{6±8}{2}
The opposite of -6 is 6.
x=\frac{14}{2}
Now solve the equation x=\frac{6±8}{2} when ± is plus. Add 6 to 8.
x=7
Divide 14 by 2.
x=-\frac{2}{2}
Now solve the equation x=\frac{6±8}{2} when ± is minus. Subtract 8 from 6.
x=-1
Divide -2 by 2.
x=7 x=-1
The equation is now solved.
x^{2}-6x-7=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.
x^{2}-6x-7-\left(-7\right)=-\left(-7\right)
Add 7 to both sides of the equation.
x^{2}-6x=-\left(-7\right)
Subtracting -7 from itself leaves 0.
x^{2}-6x=7
Subtract -7 from 0.
x^{2}-6x+\left(-3\right)^{2}=7+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=7+9
Square -3.
x^{2}-6x+9=16
Add 7 to 9.
\left(x-3\right)^{2}=16
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{16}
Take the square root of both sides of the equation.
x-3=4 x-3=-4
Simplify.
x=7 x=-1
Add 3 to both sides of the equation.
x ^ 2 -6x -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.
r + s = 6 rs = -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 = 3 - u s = 3 + u
Two numbers r and s sum up to 6 exactly when the average of the two numbers is \frac{1}{2}*6 = 3. 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>
(3 - u) (3 + u) = -7
To solve for unknown quantity u, substitute these in the product equation rs = -7
9 - u^2 = -7
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -7-9 = -16
Simplify the expression by subtracting 9 on both sides
u^2 = 16 u = \pm\sqrt{16} = \pm 4
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =3 - 4 = -1 s = 3 + 4 = 7
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.