Answer :
we have
[tex]x^{2} -7x=-13[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]x^{2} -7x+3.5^{2}=-13+3.5^{2}[/tex]
[tex]x^{2} -7x+12.25=-13+12.25[/tex]
[tex]x^{2} -7x+12.25=-0.75[/tex]
Rewrite as perfect squares
[tex](x-3.5)^{2}=-0.75[/tex]
remember that
[tex]i=\sqrt{-1}[/tex]
Square root both sides
[tex](x-3.5)=(+/-)\sqrt{-0.75}[/tex]
[tex](x-3.5)=(+/-)\sqrt{-1*0.75}[/tex]
[tex](x-3.5)=(+/-)\sqrt{0.75}i[/tex]
[tex]x1=3.5+\sqrt{0.75}i[/tex]
[tex]x2=3.5-\sqrt{0.75}i[/tex]
Simplify
[tex]\sqrt{0.75}=\frac{\sqrt{3}}{2}[/tex]
[tex]x1=3.5+\frac{\sqrt{3}}{2}i[/tex]
[tex]x2=3.5-\frac{\sqrt{3}}{2}i[/tex]
therefore
the answer is
[tex]x1=3.5+\frac{\sqrt{3}}{2}i[/tex]
[tex]x2=3.5-\frac{\sqrt{3}}{2}i[/tex]
Answer: x = [tex]\frac{7±i\sqrt{3}}{2}[/tex] (Ignore the A)
Step-by-step explanation: This is step-by-step using the QF (quadratic formula)
The first step is putting it into SF (Standard Form). Keep in mind the formula for SF is [tex]ax^{2}[/tex] ± [tex]bx[/tex] ± c= 0.
You will want to move the -13 from the right side to the left by subtracting it.
x2 – 7x = –13
+13 +13 (in order to cancel out -13 you will need to use the
--------------------------- opposite operation, in this case adding it.) You will get
[tex]x^{2}[/tex] - 7x + 13= 0 Now you want to use the QF, which is [tex]\frac{-b+/- \sqrt{b^{2} -4ac} }{2a}[/tex]
a = 1 (remember that [tex]x^{2}[/tex] has an invisible 1 in front of it) b = -7 (don"t forget any negative signs with the number(s)) c = 13
[tex]x=\frac{-(-7)+/-\sqrt{(-7)^{2}-4(1)(13) } }{2(1)}[/tex] Put everything in parenthesis to make it easier.
Use PEMDAS to solve. Start with [tex](-7)^{2}[/tex]
[tex]x = \frac{-(-7)+/-\sqrt{49-4(1)(13)} }{2(1)}[/tex] Now do all the multiplication under the square root first. Then the -(-7) and lastly the 2(1).
[tex]x=\frac{7+/-\sqrt{49-52} }{2}[/tex] (1x13=13 and 4x13= 52) Now do the math in the square root.
[tex]x=\frac{7+/-\sqrt{-3} }{2}[/tex] --- [tex]x=\frac{7+/-\sqrt{-1}* \sqrt{3} }{2}[/tex] ---- [tex]x=\frac{7+/-i \sqrt{3} }{2}[/tex] There's your answer.
Remember because [tex]\sqrt{-3}[/tex] is not a real number it has to be broken down into [tex]\sqrt{-1} *\sqrt{3}[/tex]. -1 is = to i. i stands for imagiary.