Answer:
Part A) [tex]x=(+/-)\frac{3\sqrt{3}}{2}[/tex]
Part B) [tex]x=2.5[/tex] and [tex]x=-0.5[/tex]
Part C) [tex]x=-1[/tex] and [tex]x=3[/tex]
Part D) [tex]x=\frac{9+\sqrt{193}}{8}[/tex] and [tex]x=\frac{9-\sqrt{193}}{8}[/tex]
Step-by-step explanation:
Part A) we have
[tex]4x^{2}-27=0[/tex]
we know that
The square root property states that if we have an equation with a perfect square on one side and a number on the other side, then we can take the square root of both sides and add a plus or minus sign to the side with the number and solve the equation.
isolate the term that contains the squared variable
[tex]4x^{2}=27[/tex]
[tex]x^{2} =\frac{27}{4}[/tex]
take the square root of both sides
[tex]x=(+/-)\frac{\sqrt{27}}{2}[/tex]
simplify
[tex]x=(+/-)\frac{3\sqrt{3}}{2}[/tex]
Part B) we have
[tex]4x^{2}-8x-5=0[/tex]
Using the quadratic equation
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex] is
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]4x^{2}-8x-5=0[/tex]
so
[tex]a=4\\b=-8\\c=-5[/tex]
substitute in the formula
[tex]x=\frac{-(-8)(+/-)\sqrt{-8^{2}-4(4)(-5)}} {2(4)}[/tex]
[tex]x=\frac{8(+/-)\sqrt{144}}{8}[/tex]
[tex]x=\frac{8(+/-)12}{8}[/tex]
[tex]x=\frac{8(+)12}{8}=2.5[/tex]
[tex]x=\frac{8(-)12}{8}=-0.5[/tex]
Part C) we have
[tex]4x^{2}-8x-12=0[/tex]
Using Factoring
Simplify the expression first
Divide by 4 both sides
[tex]x^{2}-2x-3=0[/tex]
Find two numbers a and b such that
a+b=-2
ab=-3
Solve the system by graphing
The solution is a=1, b=-3
see the attached figure
so
[tex]4x^{2}-8x-12=4(x-1)(x+3)[/tex]
The solutions are
[tex]x=1, x=-3[/tex]
Part D) we have
[tex]4x^{2}-9x-7=0[/tex]
Solve by completing the square
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]4x^{2}-9x=7[/tex]
Factor the leading coefficient
[tex]4(x^{2}-\frac{9}{4}x)=7[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]4(x^{2}-\frac{9}{4}x+\frac{81}{64})=7+\frac{81}{16}[/tex]
[tex]4(x^{2}-\frac{9}{4}x+\frac{81}{64})=\frac{193}{16}[/tex]
Rewrite as perfect squares
[tex]4(x-\frac{9}{8})^{2}=\frac{193}{16}[/tex]
[tex](x-\frac{9}{8})^{2}=\frac{193}{64}[/tex]
take square root both sides
[tex](x-\frac{9}{8})=(+/-)\frac{\sqrt{193}}{8}[/tex]
[tex]x=\frac{9+\sqrt{193}}{8}[/tex]
[tex]x=\frac{9-\sqrt{193}}{8}[/tex]
