Exercise 43 Page 138

To determine the number of x-intercepts we will use the discriminant of the given quadratic equation. In the Quadratic Formula, b^2-4ac is the discriminant. ax^2+bx+c=0 ⇕ x=- b±sqrt(b^2-4ac)/2a If we just want to know the number of x-intercepts without graphing the equation, we only need to work with the discriminant. Let's first rewrite the given equation in standard form.

4.25x + 3 = - 3x^2

AddEqn

LHS+3x^2=RHS+3x^2

4.25x + 3 + 3x^2 = 0

CommutativePropAdd

Commutative Property of Addition

3x^2 + 4.25x + 3 = 0

Having rewritten the equation, we can now identify the values of a, b, and c. 3x^2+ 4.25x+ 3=0 Finally, let's evaluate the discriminant.

b^2-4ac

SubstituteValues

Substitute values

4.25^2-4( 3)( 3)

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Simplify

CalcPow

Calculate power

18.0625 - 4(3)(3)

MultNegPos

(- a)b = - ab

18.0625 - 12(3)

MultNegPos

(- a)b = - ab

18.0625 - 36

SubTerm

Subtract term

- 17.9375

Since the discriminant is - 17.9375, the quadratic equation has no x-intercepts.

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.