Exercise 72 Page 288

a We want to use the discriminant of the given quadratic equation to determine the number and type of solutions. In the Quadratic Formula,

b^{2} - 4 a c

is the discriminant.

a x^{2} + b x + c = 0 \Leftrightarrow x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

If we just want to know the number of solutions, and not the solutions themselves, we only need to work with the discriminant. Let's first write all terms on the left hand side of the equation.

6 x^{2} + x - 4 = 12 \Leftrightarrow 6 x^{2} + x - 16 = 0

Since the given equation is in standard form, we can identify the values of

a,

b,

and

c .

6 x^{2} + 1 x + (- 16) = 0

Finally, let's evaluate the discriminant.

b^{2} - 4 a c

SubstituteValues

Substitute values

1^{2} - 4 (6) (- 16)

▼

Simplify

BaseOne

$1^{a} = 1$

1 - 4 (6) (- 16)

MultPosNeg

$a (- b) = - a \cdot b$

1 - 4 (- 96)

MultNegNegOnePar

$- a (- b) = a \cdot b$

1 + 384

AddTerms

Add terms

385

The discriminant is

385 .

b Since the discriminant is

385,

greater than zero, the quadratic equation has two real roots.

Extra

Further information

If the discriminant is greater than zero, the equation will have two real solutions. If it is equal to zero, the equation will have one real solution. Finally, if the discriminant is less than zero, the equation will have no real solutions.

Subchapter links

Check Your Understanding p.285

Practice and Problem Solving p.286-287

H.O.T. Problems p.287

Standardized Test Practice p.288

Spiral Review p.288

Skills Review p.288