Factor
\left(t-10\right)\left(t-7\right)
Evaluate
\left(t-10\right)\left(t-7\right)
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a+b=-17 ab=1\times 70=70
Factor the expression by grouping. First, the expression needs to be rewritten as t^{2}+at+bt+70. To find a and b, set up a system to be solved.
-1,-70 -2,-35 -5,-14 -7,-10
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 70.
-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17
Calculate the sum for each pair.
a=-10 b=-7
The solution is the pair that gives sum -17.
\left(t^{2}-10t\right)+\left(-7t+70\right)
Rewrite t^{2}-17t+70 as \left(t^{2}-10t\right)+\left(-7t+70\right).
t\left(t-10\right)-7\left(t-10\right)
Factor out t in the first and -7 in the second group.
\left(t-10\right)\left(t-7\right)
Factor out common term t-10 by using distributive property.
t^{2}-17t+70=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.
t=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 70}}{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.
t=\frac{-\left(-17\right)±\sqrt{289-4\times 70}}{2}
Square -17.
t=\frac{-\left(-17\right)±\sqrt{289-280}}{2}
Multiply -4 times 70.
t=\frac{-\left(-17\right)±\sqrt{9}}{2}
Add 289 to -280.
t=\frac{-\left(-17\right)±3}{2}
Take the square root of 9.
t=\frac{17±3}{2}
The opposite of -17 is 17.
t=\frac{20}{2}
Now solve the equation t=\frac{17±3}{2} when ± is plus. Add 17 to 3.
t=10
Divide 20 by 2.
t=\frac{14}{2}
Now solve the equation t=\frac{17±3}{2} when ± is minus. Subtract 3 from 17.
t=7
Divide 14 by 2.
t^{2}-17t+70=\left(t-10\right)\left(t-7\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and 7 for x_{2}.
x ^ 2 -17x +70 = 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 = 17 rs = 70
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{17}{2} - u s = \frac{17}{2} + u
Two numbers r and s sum up to 17 exactly when the average of the two numbers is \frac{1}{2}*17 = \frac{17}{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{17}{2} - u) (\frac{17}{2} + u) = 70
To solve for unknown quantity u, substitute these in the product equation rs = 70
\frac{289}{4} - u^2 = 70
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 70-\frac{289}{4} = -\frac{9}{4}
Simplify the expression by subtracting \frac{289}{4} on both sides
u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{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{17}{2} - \frac{3}{2} = 7 s = \frac{17}{2} + \frac{3}{2} = 10
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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