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Question: How do you solve using the completing the square method \( {x^2} - 30x = - 125 \) ?...

How do you solve using the completing the square method x2−30x=−125{x^2} - 30x = - 125 ?

Explanation

Solution

According to the question we have to solve the given quadratic expression x2−30x=−125{x^2} - 30x = - 125 which is as mentioned in the question with the help of completing the square. So, to obtain the roots/zeros of the given quadratic expression with the help of the completing square method first of all we have to divide the whole quadratic expression with the coefficient of x2{x^2} but according to the given expression the coefficient of x2{x^2} is 1 so on dividing the whole expression with 1 there will be no change in the equation.
Now, to obtain the given quadratic expression in the form of a complete square we have to divide the coefficient of x with 2 and then we have to add and subtract the square of that in the both sides of the expression.
Now, we have to right the left hand side as a perfect square and simply the right hand side.
Now, we have to take the square root in the both hand side of the given expression.

Complete step by step answer:
Step 1: First of all we have to divide the whole quadratic expression with the coefficient of x2{x^2} but according to the given expression the coefficient of x2{x^2} is 1 so on dividing the whole expression with 1 there will be no change in the equation.
Step 2: Now, to obtain the given quadratic expression in the form of complete square we have to divide the coefficient of x with 2 and then we have to add and subtract the square of that in the both sides of the expression. Hence,
⇒x2−30x+(−302)2=−125+(−302)2\Rightarrow {x^2} - 30x + {\left( { - \dfrac{{30}}{2}} \right)^2} = - 125 + {\left( { - \dfrac{{30}}{2}} \right)^2}
On solving the expression as obtained just above,
⇒x2−30x+9004=−125+9004 ⇒x2−30x+9004=−500+9004 ⇒x2−30x+9004=100  \Rightarrow {x^2} - 30x + \dfrac{{900}}{4} = - 125 + \dfrac{{900}}{4} \\\ \Rightarrow {x^2} - 30x + \dfrac{{900}}{4} = \dfrac{{ - 500 + 900}}{4} \\\ \Rightarrow {x^2} - 30x + \dfrac{{900}}{4} = 100 \\\
Step 3: Now, we have to right the left hand side as a perfect square and simply right hand side. Hence,
⇒(x−15)2=100\Rightarrow {(x - 15)^2} = 100
Step 4: Now, we have to take the square root in the both hand side of the given expression. Hence,
⇒(x−15)=100 ⇒(x−15)=±10  \Rightarrow (x - 15) = \sqrt {100} \\\ \Rightarrow (x - 15) = \pm 10 \\\
Step 5: Now, we just have to solve the expression as obtained in the solution step 4 by adding and subtracting the term of the expression. Hence,
⇒x−15=10 ⇒x=10+15 ⇒x=25  \Rightarrow x - 15 = 10 \\\ \Rightarrow x = 10 + 15 \\\ \Rightarrow x = 25 \\\
And,
⇒x−15=−10 ⇒x=−10+15 ⇒x=5  \Rightarrow x - 15 = - 10 \\\ \Rightarrow x = - 10 + 15 \\\ \Rightarrow x = 5 \\\

Hence, we have determined the solution or the roots/zeros of the given quadratic expression which are x=5,25x = 5,25 .

Note: It is necessary that we have to divide the whole expression by dividing the coefficient of x2{x^2} so that we can easily determine the complete square of the given quadratic expression.
It is necessary that we have to take the coefficient of x then we have to half the obtained coefficient, then we have to square it and add it to the both sides of the expression.