Use factoring to solve. (x-3)(x+8)=-30?

Question

Check by substitution, or by using a graphing utility and identifying x-intercepts.

Answer 1

 This equation  is a quadratic equation. they have 2 roots  x and y

(x - 3)(x + 8) = -30

Substitute  x = -3 then (-3 - 3)(-3 + 8) = -30

(-6)(5) = -30

-30 = -30

if  the first root x = -3 then x + 3 = 0

x = -3

x / -3 = 1.

Substitute y = -2 then (-2 - 3)(-2 + 8) = -30

-5(6) = -30

-30 = -30

If  the second root y = -2 then y + 2 = 0

y = -2

y / -2 = 1.

Answer 2

 This equation  is a quadratic equation. they have 2 roots  x and y

(x - 3)(x + 8) = -30

Substitute  x = -3 then (-3 - 3)(-3 + 8) = -30

(-6)(5) = -30

-30 = -30

if  the first root x = -3 then x + 3 = 0

x = -3

x / -3 = 1.

Substitute y = -2 then (-2 - 3)(-2 + 8) = -30

-5(6) = -30

-30 = -30

If  the second root y = -2 then y + 2 = 0

y = -2

y / -2 = 1

(x + 8)(x - 3) = -30

Simplify

x²- 3x + 8x  - 24 = - 30

x² + 5x -24 = -30

Add 24 to each side

x² + 5x = -6

x² + 5x + 6 = 0

x² + 3x + 2x + 6 = 0

x(x + 3) + 2(x + 3) = 0

(x + 3)(x + 2) = 0

x + 3 = 0   and x + 2 = 0

x = - 3 and x = -2.

If x + 3 is a factor then x = -3 is a root

if x + 2 is a factor then x = -2 is a another root.

Answer 3

The equation (x - 3)(x + 8) = - 30

Use factoring to solve.

• Simplify (x - 3)(x + 8) = - 30

x²- 3x + 8x  - 24 = - 30

x² + 5x - 24 = - 30

x² + 5x - 24  + 30 = 0

x² + 5x + 6 = 0

• Now factorise the equation x² + 5x  + 6 = 0

Multiply first term x² and last term 6 = 6x²

The correct pair of the terms 2x and 3x multiply to 6x² and add to 5x.

Replace the middle term 5x with  2x + 3x.

x² + 2x + 3x + 6 = 0

Group the terms into two pairs.

(x² + 2x) + ( 3x + 6) = 0

Factor out x from the first group  and factor out 3 from the second group.

x( x + 2 ) + 3( x + 2) = 0

Factor out common term x + 2.

(x + 2) (x + 3) = 0

x + 2 = 0 and x + 3 = 0

x = - 2 and x = - 3

Solution of (x - 3)(x + 8) = - 30 are x = - 2 and - 3.

Check by substitution.

Substitute  x = - 2 in (x - 3)(x + 8) = - 30

Above statement is true.

 Substitute  x = - 3 in (x - 3)(x + 8) = - 30

Above statement is true.

Answer 4

The equation (x - 3)(x + 8) = - 30

x² +5x + 6 = 0

The graph of quadratic equation is a parabola.

Graph of the parabola in intercept form :

The intecept form of quadratic equation(parabola equation) is y = a(x - p)(x - q), where p and q are x - intercepts, and

• The role of ' a '.

1. If a > 0, then the parabola opens upwards.
2. If a < 0, then the parabola opens downwards.

The function is f(x) = y =  x² +5x + 6.

Write the parabola equation in intercept form.

Factor the equation x² + 5x + 6 = 0.

x² + 2x + 3x + 6 = 0

x( x + 2 ) + 3( x + 2) = 0

(x + 2) (x + 3) = 0

y = (x +2)(x +3)

y = 1(x + 2)(x + 3)

y = 1[x - (- 2)][(x - (- 3)]

Compare it to y = a(x - p)(x - q)

a = 1 > 0 (positive), so the graph of function opens upwards and has a minimum value.

The axis of symmetry x = (p + q)/2.

x = (- 2 - 3)/2 = - 5/2

x - intercepts p and q are - 2 and - 3.

y - intercept is a(- p)(- q) = 1[-(-2))(-(-3)]

= 1(2 )( 3) = 6

y - intercept is (0,6)

• The y - intercept is (0, 6), so the point paired with it on other side of the axis of symmetry is (-5/2, 6) and has the same y - value.
  • Since, The axis of symmetry devides the parabola into two equal parts.So, if there is a point (0, 6) on one side, there is a corresponding point on other side that is the same distance from the axis of symmetry and has the same y - value.
  • The distance between the points (0, 6) and (-5/2,6) = 2.5 = The distance between (-5/2, 6) and the point paired with it on other side of the axis of symmetry and has the same y - value.
  • Therefore, The point paired with it on other side of the axis of symmetry is (- 5, 6).

• The vertex (x, y) = (x, f(x))

= (- 5/2, f(- 5/2))

The vertex (x, y) = (- 5/2, -1/4).

Graph :

1.Draw a coordinate plane.

2.Plot the coordinate points.

3.Then sketch the graph, connecting the points with a smooth curve.

From the graph x intercepts are -2 and - 3.