Part 1) x=3
Part 2) x = −1.11 and x = 1.11
Part 3) 105
Part 4) a = −6, b = 9, c = −7
Part 5) x equals 5 plus or minus the square root of 33, all over 2
Part 6) In the procedure
Part 7) ![-0.55](/tpl/images/1293/6569/c314d.png)
Part 8) The denominator is 2
Part 9) a = −6, b = −8, c = 12
Step-by-step explanation:
we know that
The formula to solve a quadratic equation of the form
is equal to
![x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}](/tpl/images/1293/6569/691f6.png)
Part 1)
in this problem we have
so
![a=1\\b=-6\\c=9](/tpl/images/1293/6569/1b6cc.png)
substitute in the formula
![x=\frac{-(-6)(+/-)\sqrt{-6^{2}-4(1)(9)}} {2(1)}](/tpl/images/1293/6569/f9e99.png)
![x=\frac{6(+/-)\sqrt{0}} {2}](/tpl/images/1293/6569/7d91a.png)
![x=\frac{6} {2}=3](/tpl/images/1293/6569/a5523.png)
Part 2) in this problem we have
so
![a=49\\b=0\\c=-60](/tpl/images/1293/6569/cf2f3.png)
substitute in the formula
![x=\frac{0(+/-)\sqrt{0^{2}-4(49)(-60)}} {2(49)}](/tpl/images/1293/6569/e14e7.png)
![x=\frac{0(+/-)\sqrt{11,760}} {98}](/tpl/images/1293/6569/0067f.png)
![x=(+/-)1.11](/tpl/images/1293/6569/b7902.png)
Part 3) When the solution of x2 − 9x − 6 is expressed as 9 plus or minus the square root of r, all over 2, what is the value of r?
in this problem we have
so
![a=1\\b=-9\\c=-6](/tpl/images/1293/6569/9247b.png)
substitute in the formula
![x=\frac{-(-9)(+/-)\sqrt{-9^{2}-4(1)(-6)}} {2(1)}](/tpl/images/1293/6569/6f5d0.png)
![x=\frac{9(+/-)\sqrt{105}} {2}](/tpl/images/1293/6569/c53ab.png)
therefore
![r=105](/tpl/images/1293/6569/67ab8.png)
Part 4) What are the values a, b, and c in the following quadratic equation?
−6x2 = −9x + 7
in this problem we have
so
![a=-6\\b=9\\c=-7](/tpl/images/1293/6569/eaf6b.png)
Part 5) Use the quadratic formula to find the exact solutions of x2 − 5x − 2 = 0.
In this problem we have
so
![a=1\\b=-5\\c=-2](/tpl/images/1293/6569/6bee7.png)
substitute in the formula
![x=\frac{-(-5)(+/-)\sqrt{-5^{2}-4(1)(-2)}} {2(1)}](/tpl/images/1293/6569/32273.png)
![x=\frac{5(+/-)\sqrt{33}} {2}](/tpl/images/1293/6569/7f9f9.png)
therefore
x equals 5 plus or minus the square root of 33, all over 2
Part 6) Quadratic Formula proof
we have
Divide both sides by a
Complete the square
![x^{2} +\frac{b}{a}x+\frac{b^{2}}{4a^{2}} =-\frac{c}{a}+\frac{b^{2}}{4a^{2}}](/tpl/images/1293/6569/9d321.png)
Rewrite the perfect square trinomial on the left side of the equation as a binomial squared
![(x+\frac{b}{2a})^{2}=-\frac{4ac}{a^{2}}+\frac{b^{2}}{4a^{2}}](/tpl/images/1293/6569/3841a.png)
Find a common denominator on the right side of the equation
![(x+\frac{b}{2a})^{2}=\frac{b^{2}-4ac}{4a^{2}}](/tpl/images/1293/6569/9a49b.png)
Take the square root of both sides of the equation
![(x+\frac{b}{2a})=(+/-)\sqrt{\frac{b^{2}-4ac}{4a^{2}}}](/tpl/images/1293/6569/05d88.png)
Simplify the right side of the equation
![(x+\frac{b}{2a})=(+/-)\frac{\sqrt{b^{2}-4ac}}{2a}](/tpl/images/1293/6569/89609.png)
Subtract the quantity b over 2 times a from both sides of the equation
![x=-\frac{b}{2a}(+/-)\frac{\sqrt{b^{2}-4ac}}{2a}](/tpl/images/1293/6569/8fa30.png)
![x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}](/tpl/images/1293/6569/691f6.png)
Part 7) in this problem we have
so
![a=3\\b=45\\c=24](/tpl/images/1293/6569/76125.png)
substitute in the formula
![x=\frac{-(45)(+/-)\sqrt{45^{2}-4(3)(24)}} {2(3)}](/tpl/images/1293/6569/a6842.png)
![x=\frac{-(45)(+/-)\sqrt{1,737}} {6}](/tpl/images/1293/6569/f6394.png)
![x1=\frac{-(45)(+)\sqrt{1,737}} {6}=-0.55](/tpl/images/1293/6569/5d505.png)
![x2=\frac{-(45)(-)\sqrt{1,737}} {6}=-14.45](/tpl/images/1293/6569/bd628.png)
therefore
The other solution is
![-0.55](/tpl/images/1293/6569/c314d.png)
Part 8) in this problem we have
so
![a=2\\b=-8\\c=7](/tpl/images/1293/6569/3cdec.png)
substitute in the formula
![x=\frac{-(-8)(+/-)\sqrt{-8^{2}-4(2)(7)}} {2(2)}](/tpl/images/1293/6569/5e362.png)
![x=\frac{8(+/-)\sqrt{8}} {4}](/tpl/images/1293/6569/2ebf6.png)
![x=\frac{8(+/-)2\sqrt{2}} {4}](/tpl/images/1293/6569/8b12b.png)
![x=\frac{4(+/-)\sqrt{2}} {2}](/tpl/images/1293/6569/1cd66.png)
therefore
The denominator is 2
Part 9) What are the values a, b, and c in the following quadratic equation?
−6x2 − 8x + 12
in this problem we have
so
![a=-6\\b=-8\\c=12](/tpl/images/1293/6569/a7ed9.png)