Solving Quadratic Equations - GMAT Quantitative

Substitute , and, subsequently, , and solve the resulting quadratic equation.

We can rewrite the quadratic expression as , where the question marks are replaced with integers whose product is and whose sum is 9; the integers are :

Set each factor to zero and solve for ; then substitute back and solve for :

or

, which has no real solution.

Therefore, the solution set is

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:

or

The solution set is

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:

or

The solution set is .

Subsitute in the defintion, and set it equal to 21:

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

or

The solution set is