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Question: How do you factor \[10{x^2} - 7x - 12\]?...

How do you factor 10x2−7x−1210{x^2} - 7x - 12?

Explanation

Solution

This problem deals with solving a quadratic equation. Here, given a quadratic equation expression, we have to simplify the expression and make it into a standard form of quadratic equation. If the quadratic equation is in the form of ax2+bx+c=0a{x^2} + bx + c = 0, then we know that the roots of this quadratic equation are given by :
⇒x=−b±b2−4ac2a\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}

Complete step-by-step answer:
The given expression is a quadratic expression but not a quadratic equation. We have to convert the expression into an equation and then solve the quadratic equation.
The given expression is 10x2−7x−1210{x^2} - 7x - 12, consider it as given below:
⇒10x2−7x−12\Rightarrow 10{x^2} - 7x - 12
Now this is in the standard form of a quadratic expression, now comparing the coefficients a,ba,b and cc:
⇒a=10,b=−7\Rightarrow a = 10,b = - 7 and c=−12c = - 12
Now applying the formula to find the value of the roots of xx, as given below:
⇒x=−b±b2−4ac2a\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
Substituting the values of a,ba,b and cc in the above formula:
⇒x=−(−7)±(−7)2−4(10)(−12)2(10)\Rightarrow x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( {10} \right)\left( { - 12} \right)} }}{{2\left( {10} \right)}}
⇒x=7±49+48020\Rightarrow x = \dfrac{{7 \pm \sqrt {49 + 480} }}{{20}}
⇒x=7±52920\Rightarrow x = \dfrac{{7 \pm \sqrt {529} }}{{20}}
We know that the square root of 529 is 23, 529=23\sqrt {529} = 23
⇒x=7±2320\Rightarrow x = \dfrac{{7 \pm 23}}{{20}}
Now considering the two cases, with plus and minus, as shown:
⇒x=7+2320;x=7−2320\Rightarrow x = \dfrac{{7 + 23}}{{20}};x = \dfrac{{7 - 23}}{{20}}
⇒x=3020;x=−1620\Rightarrow x = \dfrac{{30}}{{20}};x = \dfrac{{ - 16}}{{20}}
Hence the value of the roots are equal to :
∴x=32;x=−45\therefore x = \dfrac{3}{2};x = \dfrac{{ - 4}}{5}
Now factoring the given expression as 10x2−7x−12=(x−32)(x+45)10{x^2} - 7x - 12 = \left( {x - \dfrac{3}{2}} \right)\left( {x + \dfrac{4}{5}} \right)
∴10x2−7x−12=(2x−3)(5x+4)\therefore 10{x^2} - 7x - 12 = \left( {2x - 3} \right)\left( {5x + 4} \right)

Final Answer: The factors of the expression 10x2−7x−1210{x^2} - 7x - 12 are (2x−3)\left( {2x - 3} \right)and (5x+4)\left( {5x + 4} \right).

Note:
Please note that this problem can also be done either by the method of completing the square or just factoring and solving the quadratic equation. To solve ax2+bx+c=0a{x^2} + bx + c = 0 by completing the square: transform the equation so that the constant term, cc is alone on the right side. But here we are adding and subtracting some terms in order to factor.