Question
Completing the Square
Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary.
1. 22-8x+16=1
2. x2 + 4x +4 = 1
-
3. x2 + 12x + 36 = 25
5. x2 + 4x + 4 =2
7.22-68+9=7
4. 4x2 - 4x + 1 = 9
6. x2 - 2x+1 = 5
8. x2 + 16x + 64 = 15
/b-Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect squar
10. x7 - 14x +c
9. x2 + 10x + c
11.22+24x+C
13. x2 - 9x+ c
12. x2 + 5x +C
14.22-8+C
Solve each equation by completing the square.
15. x2 - 13x + 36=0
16. x2 + 3x = 0
17. x2+x - 6=0
18.22-48-13=0
19. 2x2 + 7x - 4 = 0
20.3x2 + 2x - 1 = 0
21. x2 + 3x - 6 = 0
22. x2-x - 3 = 0
23.22=-11
24. x2 - 2x +4 = 0
Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary.
1. 22-8x+16=1
2. x2 + 4x +4 = 1
-
3. x2 + 12x + 36 = 25
5. x2 + 4x + 4 =2
7.22-68+9=7
4. 4x2 - 4x + 1 = 9
6. x2 - 2x+1 = 5
8. x2 + 16x + 64 = 15
/b-Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect squar
10. x7 - 14x +c
9. x2 + 10x + c
11.22+24x+C
13. x2 - 9x+ c
12. x2 + 5x +C
14.22-8+C
Solve each equation by completing the square.
15. x2 - 13x + 36=0
16. x2 + 3x = 0
17. x2+x - 6=0
18.22-48-13=0
19. 2x2 + 7x - 4 = 0
20.3x2 + 2x - 1 = 0
21. x2 + 3x - 6 = 0
22. x2-x - 3 = 0
23.22=-11
24. x2 - 2x +4 = 0
Answers
Answered by
GPT-4o mini
Let's go through your equations step by step, solving the first set using the Square Root Property, then finding the value of \(c\) to make each trinomial a perfect square, and finally solving the equations by completing the square.
### Using the Square Root Property
**1.** \(22 - 8x + 16 = 1\)
Combine terms: \(38 - 8x = 1\)
Subtract 38: \(-8x = -37\)
Divide by -8:
\[ x = \frac{37}{8} \approx 4.63 \]
**2.** \(x^2 + 4x + 4 = 1\)
Turn it into standard form: \(x^2 + 4x + 3 = 0\)
Factor: \((x + 3)(x + 1) = 0\)
Solutions: \(x = -3, -1\)
**3.** \(x^2 + 12x + 36 = 25\)
Standard form: \(x^2 + 12x + 11 = 0\)
Factor: \((x + 11)(x + 1) = 0\)
Solutions: \(x = -11, -1\)
**4.** \(4x^2 - 4x + 1 = 9\)
Standard form: \(4x^2 - 4x - 8 = 0\)
Divide everything by 4: \(x^2 - x - 2 = 0\)
Factor: \((x - 2)(x + 1) = 0\)
Solutions: \(x = 2, -1\)
**5.** \(x^2 + 4x + 4 = 2\)
Standard form: \(x^2 + 4x + 2 = 0\)
Using the square root property:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(x = \frac{-4 \pm \sqrt{16 - 8}}{2} = \frac{-4 \pm \sqrt{8}}{2} = \frac{-4 \pm 2\sqrt{2}}{2}\)
Solutions: \(x = -2 - \sqrt{2}, -2 + \sqrt{2}\)
**6.** \(x^2 - 2x + 1 = 5\)
Standard form: \(x^2 - 2x - 4 = 0\)
Using the square root property:
\(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)}\)
\[x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5} \]
**7.** \(22 - 6x + 9 = 7\)
Combine: \(31 - 6x = 7\)
Standard form: \(-6x = -24\)
\[ x = 4 \]
**8.** \(x^2 + 16x + 64 = 15\)
Standard form: \(x^2 + 16x + 49 = 0\)
Factor: \((x + 7)(x + 7) = 0\)
Solution: \(x = -7\)
### Find the value of c for perfect squares
**9.** \(x^2 + 10x + c\)
C is \((\frac{10}{2})^2 = 25\)
Perfect square: \((x + 5)^2\)
**10.** \(x^2 - 14x + c\)
C is \((\frac{-14}{2})^2 = 49\)
Perfect square: \((x - 7)^2\)
**11.** \(x^2 + 24x + C\)
C is \((\frac{24}{2})^2 = 144\)
Perfect square: \((x + 12)^2\)
**12.** \(x^2 + 5x + C\)
C is \((\frac{5}{2})^2 = \frac{25}{4}\)
Perfect square: \((x + \frac{5}{2})^2\)
**13.** \(x^2 - 9x + c\)
C is \((\frac{-9}{2})^2 = \frac{81}{4}\)
Perfect square: \((x - \frac{9}{2})^2\)
**14.** \(x^2 - 8 + C\)
C is \((\frac{-8}{2})^2 = 16\)
Perfect square: \((x - 4)^2\)
### Completing the square for the other equations
**15.** \(x^2 - 13x + 36 = 0\)
Move 36: \(x^2 - 13x = -36\)
Complete square: \(x^2 - 13x + \left(\frac{13}{2}\right)^2 = -36 + \left(\frac{13}{2}\right)^2\)
\[ \left(x - \frac{13}{2}\right)^2 = -36 + \frac{169}{4} \]
Solve for \(x\).
**16.** \(x^2 + 3x = 0\)
Factor: \(x(x + 3) = 0\)
Solutions: \(x = 0, -3\)
**17.** \(x^2 + x - 6 = 0\)
Complete square:
Move -6: \(x^2 + x = 6\)
Complete: \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
\[\left(x + \frac{1}{2}\right)^2 = 6 + \frac{1}{4}\]
Solve for \(x\).
**18.** \(x^2 - 48 - 13 = 0\)
Complete: \(x^2 = 61\)
Solutions: \(x = \sqrt{61}, -\sqrt{61}\)
**19.** \(2x^2 + 7x - 4 = 0\)
Using quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Solve for \(x\).
**20.** \(3x^2 + 2x - 1 = 0\)
Using quadratic formula:
Solutions: \(x = \frac{-2 \pm \sqrt{2^2 - 4(3)(-1)}}{2(3)}\)
**21.** \(x^2 + 3x - 6 = 0\)
Complete square:
Move -6: \(x^2 + 3x = 6\)
Complete: Solutions here.
**22.** \(x^2 - x - 3 = 0\)
Complete squares here.
**23.** \(x^2 = -11\)
Complex solutions: \(x = i\sqrt{11}, -i\sqrt{11}\)
**24.** \(x^2 - 2x + 4 = 0\)
Complete: Complex solutions from formula.
Please substitute values and articulate your findings in practice! For explicit answers or further clarifications on certain steps, feel free to ask!
### Using the Square Root Property
**1.** \(22 - 8x + 16 = 1\)
Combine terms: \(38 - 8x = 1\)
Subtract 38: \(-8x = -37\)
Divide by -8:
\[ x = \frac{37}{8} \approx 4.63 \]
**2.** \(x^2 + 4x + 4 = 1\)
Turn it into standard form: \(x^2 + 4x + 3 = 0\)
Factor: \((x + 3)(x + 1) = 0\)
Solutions: \(x = -3, -1\)
**3.** \(x^2 + 12x + 36 = 25\)
Standard form: \(x^2 + 12x + 11 = 0\)
Factor: \((x + 11)(x + 1) = 0\)
Solutions: \(x = -11, -1\)
**4.** \(4x^2 - 4x + 1 = 9\)
Standard form: \(4x^2 - 4x - 8 = 0\)
Divide everything by 4: \(x^2 - x - 2 = 0\)
Factor: \((x - 2)(x + 1) = 0\)
Solutions: \(x = 2, -1\)
**5.** \(x^2 + 4x + 4 = 2\)
Standard form: \(x^2 + 4x + 2 = 0\)
Using the square root property:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(x = \frac{-4 \pm \sqrt{16 - 8}}{2} = \frac{-4 \pm \sqrt{8}}{2} = \frac{-4 \pm 2\sqrt{2}}{2}\)
Solutions: \(x = -2 - \sqrt{2}, -2 + \sqrt{2}\)
**6.** \(x^2 - 2x + 1 = 5\)
Standard form: \(x^2 - 2x - 4 = 0\)
Using the square root property:
\(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)}\)
\[x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5} \]
**7.** \(22 - 6x + 9 = 7\)
Combine: \(31 - 6x = 7\)
Standard form: \(-6x = -24\)
\[ x = 4 \]
**8.** \(x^2 + 16x + 64 = 15\)
Standard form: \(x^2 + 16x + 49 = 0\)
Factor: \((x + 7)(x + 7) = 0\)
Solution: \(x = -7\)
### Find the value of c for perfect squares
**9.** \(x^2 + 10x + c\)
C is \((\frac{10}{2})^2 = 25\)
Perfect square: \((x + 5)^2\)
**10.** \(x^2 - 14x + c\)
C is \((\frac{-14}{2})^2 = 49\)
Perfect square: \((x - 7)^2\)
**11.** \(x^2 + 24x + C\)
C is \((\frac{24}{2})^2 = 144\)
Perfect square: \((x + 12)^2\)
**12.** \(x^2 + 5x + C\)
C is \((\frac{5}{2})^2 = \frac{25}{4}\)
Perfect square: \((x + \frac{5}{2})^2\)
**13.** \(x^2 - 9x + c\)
C is \((\frac{-9}{2})^2 = \frac{81}{4}\)
Perfect square: \((x - \frac{9}{2})^2\)
**14.** \(x^2 - 8 + C\)
C is \((\frac{-8}{2})^2 = 16\)
Perfect square: \((x - 4)^2\)
### Completing the square for the other equations
**15.** \(x^2 - 13x + 36 = 0\)
Move 36: \(x^2 - 13x = -36\)
Complete square: \(x^2 - 13x + \left(\frac{13}{2}\right)^2 = -36 + \left(\frac{13}{2}\right)^2\)
\[ \left(x - \frac{13}{2}\right)^2 = -36 + \frac{169}{4} \]
Solve for \(x\).
**16.** \(x^2 + 3x = 0\)
Factor: \(x(x + 3) = 0\)
Solutions: \(x = 0, -3\)
**17.** \(x^2 + x - 6 = 0\)
Complete square:
Move -6: \(x^2 + x = 6\)
Complete: \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
\[\left(x + \frac{1}{2}\right)^2 = 6 + \frac{1}{4}\]
Solve for \(x\).
**18.** \(x^2 - 48 - 13 = 0\)
Complete: \(x^2 = 61\)
Solutions: \(x = \sqrt{61}, -\sqrt{61}\)
**19.** \(2x^2 + 7x - 4 = 0\)
Using quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Solve for \(x\).
**20.** \(3x^2 + 2x - 1 = 0\)
Using quadratic formula:
Solutions: \(x = \frac{-2 \pm \sqrt{2^2 - 4(3)(-1)}}{2(3)}\)
**21.** \(x^2 + 3x - 6 = 0\)
Complete square:
Move -6: \(x^2 + 3x = 6\)
Complete: Solutions here.
**22.** \(x^2 - x - 3 = 0\)
Complete squares here.
**23.** \(x^2 = -11\)
Complex solutions: \(x = i\sqrt{11}, -i\sqrt{11}\)
**24.** \(x^2 - 2x + 4 = 0\)
Complete: Complex solutions from formula.
Please substitute values and articulate your findings in practice! For explicit answers or further clarifications on certain steps, feel free to ask!