Chapter Fourteen — Quadratic Equations and Inequalities

The Humongous Book of Algebra Problems

312

Solve the equation using the quadratic formula.

Reduce the fraction to lowest terms.

Squaring both sides of an equation might introduce false answers, so verify the

solutions by substituting them into the original equation.

Neither nor satisﬁes the equation, so the equation

has no real number solution.

Applying the Discriminant

What b

2

– 4ac tells you about an equation

14.29 The discriminant of the equation ax

2

+ bx + c = 0 is deﬁned as the value

b

2

– 4ac. What conclusions can be drawn about the real roots of a quadratic

equation based on the sign of its discriminant?

According to the fundamental theorem of algebra, all quadratic equations have

two roots, because all quadratic equations have degree two. The discriminant is

calculated to predict how many (if any) of those roots will be real numbers.

Getting

to the last

steps from the

steps above them

is quite advanced,

so don’t worry about

simplifying something

like

.

Instead, type the

expressions into a

calculator. The

decimal you get for

the left side won’t

equal the decimal you

get for the right side,

and that’s enough

to show that the

equation is false.

“Roots”

is another

words for

“solutions,” so the

book’s asking what

you can predict

about the real number

solutions of an equation.

Don’t get these roots

confused with radicals;

square ROOTS

and the ROOTS

of an equation

mean totally

different

things.

Chapter Fourteen — Quadratic Equations and Inequalities

The Humongous Book of Algebra Problems

313

A positive discriminant indicates that the equation has two unique real roots.

In other words, there are two different, real number solutions to the quadratic.

The solutions may be rational (which means that the equation can be solved

by factoring) or irrational (which means that the equation must be solved by

completing the square or the quadratic equation), but both solutions are real

numbers.

A zero discriminant is an indication of a double root. The quadratic still has

two solutions, but the solutions are equal. For instance, the quadratic equation

(x + 5)(x + 5) = 0 has two roots: x = –5 and x = –5. Because a single real root is

repeated, x = –5 is described as a double root.

If a quadratic equation has a negative discriminant, there are no real number

solutions. That does not mean the equation has no solutions, but instead that

the solutions are complex numbers.

14.30 Given the equation 3x

2

+ 7x + 1 = 0, use the discriminant to predict how many

of the roots (if any) are real numbers, then calculate the roots to verify the

prediction made by the discriminant.

The equation has form ax

2

+ bx + c = 0, so substitute a = 3, b = 7, and c = 1 into

the discriminant formula.

According to Problem 14.29, the equation 3x

2

+ 7x + 1 = 0 has two positive

real roots because the discriminant is positive (37 > 0). Calculate the roots by

applying the quadratic formula.

The solution to the equation is or .

And they’ll

contain the

imaginary number

.

In case you

didn’t recognize

it, the discriminant

b

2

– 4ac is the part

of the quadratic

formula that’s inside

the radical. When you

plug everything into the

quadratic formula,

there’s no need to

calculate b

2

– 4ac

again—just plug in

the discriminant, 37.

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