SOLVING QUADRATICS WORKSHEET Solve using the Square Root Method 1) -5x – 4 =-504 2) x = 81 Solve by Factoring: 3) 5x” 16x + 3 = 0 4) x + 5x+ 6 = 0 5) x – 16 = 0 6) x _2x =-1 Solve by Completing the Square: 7) 3x? + 18x + 10 = 0 8) x – 2x – 15 = 0 Solve using the Quadratic Formula: 9) 4x” -X-5=0 10) 21″ + 3x – 3 =0
The Correct Answer and Explanation is:
Let’s break down each problem and solve them one by one.
Square Root Method
1) -5x – 4 = -504
Step 1: Isolate the term with the variable. −5x=−504+4 ⟹ −5x=−500-5x = -504 + 4 \implies -5x = -500−5x=−504+4⟹−5x=−500
Step 2: Divide by -5. x=−500−5 ⟹ x=100x = \frac{-500}{-5} \implies x = 100x=−5−500⟹x=100
Answer: x=100x = 100x=100
2) x = 81
This is already in the form x=81x = 81x=81.
Answer: x=81x = 81x=81
Factoring
3) 5x² – 16x + 3 = 0
To factor, we look for two numbers that multiply to 5×3=155 \times 3 = 155×3=15 and add to −16-16−16.
The pair of numbers that work is -15 and -1. So: 5×2−15x−x+3=05x² – 15x – x + 3 = 05×2−15x−x+3=0
Now factor by grouping: 5x(x−3)−1(x−3)=05x(x – 3) – 1(x – 3) = 05x(x−3)−1(x−3)=0
Factor out (x−3)(x – 3)(x−3): (5x−1)(x−3)=0(5x – 1)(x – 3) = 0(5x−1)(x−3)=0
Set each factor equal to zero: 5x−1=0orx−3=05x – 1 = 0 \quad \text{or} \quad x – 3 = 05x−1=0orx−3=0
Solving each: x=15orx=3x = \frac{1}{5} \quad \text{or} \quad x = 3x=51orx=3
Answer: x=15x = \frac{1}{5}x=51 or x=3x = 3x=3
4) x² + 5x + 6 = 0
Look for two numbers that multiply to 6 and add to 5. These are 2 and 3. So: (x+2)(x+3)=0(x + 2)(x + 3) = 0(x+2)(x+3)=0
Set each factor equal to zero: x+2=0orx+3=0x + 2 = 0 \quad \text{or} \quad x + 3 = 0x+2=0orx+3=0
Solving each: x=−2orx=−3x = -2 \quad \text{or} \quad x = -3x=−2orx=−3
Answer: x=−2x = -2x=−2 or x=−3x = -3x=−3
5) x – 16 = 0
This is a simple linear equation: x=16x = 16x=16
Answer: x=16x = 16x=16
6) x² – 2x = -1
Move the constant to the right side: x2−2x+1=0x² – 2x + 1 = 0x2−2x+1=0
Factor: (x−1)(x−1)=0(x – 1)(x – 1) = 0(x−1)(x−1)=0
Solve: x=1x = 1x=1
Answer: x=1x = 1x=1
Completing the Square
7) 3x² + 18x + 10 = 0
Step 1: Move the constant to the other side: 3×2+18x=−103x² + 18x = -103×2+18x=−10
Step 2: Divide by 3: x2+6x=−103x² + 6x = -\frac{10}{3}x2+6x=−310
Step 3: Take half of 6, square it: (62)2=9\left(\frac{6}{2}\right)² = 9(26)2=9, and add 9 to both sides: x2+6x+9=−103+9x² + 6x + 9 = -\frac{10}{3} + 9×2+6x+9=−310+9 x2+6x+9=173x² + 6x + 9 = \frac{17}{3}x2+6x+9=317
Step 4: Factor the left side: (x+3)2=173(x + 3)² = \frac{17}{3}(x+3)2=317
Step 5: Take the square root of both sides: x+3=±173x + 3 = \pm \sqrt{\frac{17}{3}}x+3=±317
Step 6: Solve for xxx: x=−3±513x = -3 \pm \frac{\sqrt{51}}{3}x=−3±351
Answer: x=−3±513x = -3 \pm \frac{\sqrt{51}}{3}x=−3±351
8) x² – 2x – 15 = 0
Step 1: Move the constant to the other side: x2−2x=15x² – 2x = 15×2−2x=15
Step 2: Take half of -2, square it: (−22)2=1\left(\frac{-2}{2}\right)² = 1(2−2)2=1, and add 1 to both sides: x2−2x+1=15+1x² – 2x + 1 = 15 + 1×2−2x+1=15+1 (x−1)2=16(x – 1)² = 16(x−1)2=16
Step 3: Take the square root of both sides: x−1=±4x – 1 = \pm 4x−1=±4
Step 4: Solve for xxx: x=1±4x = 1 \pm 4x=1±4
So, x=5x = 5x=5 or x=−3x = -3x=−3.
Answer: x=5x = 5x=5 or x=−3x = -3x=−3
Quadratic Formula
9) 4x² – x – 5 = 0
Use the quadratic formula: x=−(−1)±(−1)2−4(4)(−5)2(4)x = \frac{-(-1) \pm \sqrt{(-1)² – 4(4)(-5)}}{2(4)}x=2(4)−(−1)±(−1)2−4(4)(−5) x=1±1+808x = \frac{1 \pm \sqrt{1 + 80}}{8}x=81±1+80 x=1±818x = \frac{1 \pm \sqrt{81}}{8}x=81±81 x=1±98x = \frac{1 \pm 9}{8}x=81±9
So, x=1+98=108=54x = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4}x=81+9=810=45 or x=1−98=−88=−1x = \frac{1 – 9}{8} = \frac{-8}{8} = -1x=81−9=8−8=−1.
Answer: x=54x = \frac{5}{4}x=45 or x=−1x = -1x=−1
10) 2x² + 3x – 3 = 0
Use the quadratic formula: x=−3±32−4(2)(−3)2(2)x = \frac{-3 \pm \sqrt{3² – 4(2)(-3)}}{2(2)}x=2(2)−3±32−4(2)(−3) x=−3±9+244x = \frac{-3 \pm \sqrt{9 + 24}}{4}x=4−3±9+24 x=−3±334x = \frac{-3 \pm \sqrt{33}}{4}x=4−3±33
Answer: x=−3±334x = \frac{-3 \pm \sqrt{33}}{4}x=4−3±33
These are the solutions for each equation, along with the necessary steps for each method. Let me know if you need further explanations or if you’d like more examples!
