Determine the roots of x^2+12x-6=7x. What does "determine the roots" even mean?

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

x^2 + 12x - 6 = 7x

determine the roots , that means that we need to calculate the functions soultion in which satisfy the equality.

First let us move 7x to the left side:

==> x^2 + 5x - 6 = 0

Now the roots would be x values in which the function equals zero.

Let us factor:

x^2 + 5x -6 = (x+6)(x-1) = 0

In order for the equality to0 hold, then :

(x+ 6 = 0     OR     x-1 = 0

==> x= -6       OR   x= 1

Then the roots are :

x1= -6 

x2= 1

krishna-agrawala's profile pic

krishna-agrawala | College Teacher | (Level 3) Valedictorian

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In a quadratic equation, that is an equation in which the variable is squared, the solutions for the values of the variable are called roots of the equation.For example consider the equation below:

ax^2 + bx + c = 0

In this equation x is the variable. The quantities represented by a, b and c are known constants.

This term root is used because the to get the value of x we need to find the value of square root of x^2. As square root of x^2 can be either +x, or -x, typically a quadratic equation has two roots.

We can find the root of the given equation by first taking all the terms of the equation on the left hand side and factorising this expression, and then equating each factor to 0.

Thus:

x^2 + 12x - 6 = 7x

==> x^2 + 12x - 7x - 6 = 0

==> x^2 + 5x - 6 = 0

==> x^2 + 6x - x - 6 = 0

==> x(x + 6)- (x + 6) = 0

==> (x + 6)(x - 1) = 0

Therefore:

x = - 6 and 1

Thus roots of give equation are -6, and 1

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

x^2+12x-6 = 7x. To determine the roots.

We shall subtract 7x and rewrite the equation:

x^2+12x-6-7x = 0

x^2+5x-6 = 0

x^2+6x -x-6 = 0

x(x+6) -1(x+6) = 0

(x+6)(x-1) = 0

(x+6)= 0  or x-1 = 0

 x= -6 or x = 1

The roots of  f(x) = 0 means the values of x which makes f(a) = 0.

x = 1 = 0 is a root of x^2+12x- 6= 7x means , if you put x= 1 in the equation , it is verified or satisfied:

 1^2+12*1-6 = 7*1. Or 

1+12-6 = 7

7 = 7.

Similarly x= -6 satisfies the equation:

(-6)^2+12*-6-6= -7*6

36 -72-6 = 7*6

-36-6 = -42

-42 = -42.

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