Exercises What is the value of the discriminant and what is the number of real solutions for each equation? $$ \begin{array}{lll} ext { 7. } x^{2}+x-42=0 & ext { 8. }-x^{2}+13 x-40=0 & ext { 9. } x^{2}+2 x+5=0 \ \begin{array}{lll} ext { 10. } x^{2}=18 x-81 & ext { 11. }-x^{2}+7 x+44=0 & ext { 12. } \frac{1}{4} x^{2}-5 x+25=0\end{array} \ \begin{array}{lll} ext { 13. } 2 x^{2}+7=5 x & ext { 14. } 4 x^{2}+25 x=21 & ext { 17. } \frac{1}{2} x^{2}+2 x+3=0\end{array} & ext { 18. } \frac{1}{6} x^{2}=2 x+18\end{array} $$ Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Exercises What is the value of the discriminant and what is the number of real solutions for each equation? $$ \begin{array}{lll}	ext { 7. } x^{2}+x-42=0 & 	ext { 8. }-x^{2}+13 x-40=0 & 	ext { 9. } x^{2}+2 x+5=0 \ \begin{array}{lll}	ext { 10. } x^{2}=18 x-81 & 	ext { 11. }-x^{2}+7 x+44=0 & 	ext { 12. } \frac{1}{4} x^{2}-5 x+25=0\end{array} \ \begin{array}{lll}	ext { 13. } 2 x^{2}+7=5 x & 	ext { 14. } 4 x^{2}+25 x=21 & 	ext { 17. } \frac{1}{2} x^{2}+2 x+3=0\end{array} & 	ext { 18. } \frac{1}{6} x^{2}=2 x+18\end{array} $$ Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Sure, let's analyze each of the given quadratic equations to determine their discriminants and the number of real solutions they possess. The discriminant $$ D $$ of a quadratic equation $$ ax^2 + bx + c = 0 $$ is calculated using the formula: $$ D = b^2 - 4ac $$ Based on the value of $$ D $$: - If $$ D > 0 $$: **Two distinct real solutions** - If $$ D = 0 $$: **One real solution (a repeated root)** - If $$ D < 0 $$: **No real solutions (two complex solutions)** Here's the breakdown for each equation: --- ### **7. $$ x^{2} + x - 42 = 0 $$** - **Coefficients:** $$ a = 1 $$, $$ b = 1 $$, $$ c = -42 $$ - **Discriminant:** $$ D = 1^2 - 4(1)(-42) = 1 + 168 = 169 $$ - **Number of Real Solutions:** **2** (since $$ D = 169 > 0 $$) --- ### **8. $$ -x^{2} + 13x - 40 = 0 $$** - **Coefficients:** $$ a = -1 $$, $$ b = 13 $$, $$ c = -40 $$ - **Discriminant:** $$ D = 13^2 - 4(-1)(-40) = 169 - 160 = 9 $$ - **Number of Real Solutions:** **2** (since $$ D = 9 > 0 $$) --- ### **9. $$ x^{2} + 2x + 5 = 0 $$** - **Coefficients:** $$ a = 1 $$, $$ b = 2 $$, $$ c = 5 $$ - **Discriminant:** $$ D = 2^2 - 4(1)(5) = 4 - 20 = -16 $$ - **Number of Real Solutions:** **No real solutions** (since $$ D = -16 < 0 $$) --- ### **10. $$ x^{2} = 18x - 81 $$** - **Rearranged Equation:** $$ x^{2} - 18x + 81 = 0 $$ - **Coefficients:** $$ a = 1 $$, $$ b = -18 $$, $$ c = 81 $$ - **Discriminant:** $$ D = (-18)^2 - 4(1)(81) = 324 - 324 = 0 $$ - **Number of Real Solutions:** **1** (since $$ D = 0 $$) --- ### **11. $$ -x^{2} + 7x + 44 = 0 $$** - **Coefficients:** $$ a = -1 $$, $$ b = 7 $$, $$ c = 44 $$ - **Discriminant:** $$ D = 7^2 - 4(-1)(44) = 49 + 176 = 225 $$ - **Number of Real Solutions:** **2** (since $$ D = 225 > 0 $$) --- ### **12. $$ \frac{1}{4}x^{2} - 5x + 25 = 0 $$** - **Coefficients:** $$ a = \frac{1}{4} $$, $$ b = -5 $$, $$ c = 25 $$ - **Discriminant:** $$ D = (-5)^2 - 4\left(\frac{1}{4}\right)(25) = 25 - 25 = 0 $$ - **Number of Real Solutions:** **1** (since $$ D = 0 $$) --- ### **13. $$ 2x^{2} + 7 = 5x $$** - **Rearranged Equation:** $$ 2x^{2} - 5x + 7 = 0 $$ - **Coefficients:** $$ a = 2 $$, $$ b = -5 $$, $$ c = 7 $$ - **Discriminant:** $$ D = (-5)^2 - 4(2)(7) = 25 - 56 = -31 $$ - **Number of Real Solutions:** **No real solutions** (since $$ D = -31 < 0 $$) --- ### **14. $$ 4x^{2} + 25x = 21 $$** - **Rearranged Equation:** $$ 4x^{2} + 25x - 21 = 0 $$ - **Coefficients:** $$ a = 4 $$, $$ b = 25 $$, $$ c = -21 $$ - **Discriminant:** $$ D = 25^2 - 4(4)(-21) = 625 + 336 = 961 $$ - **Number of Real Solutions:** **2** (since $$ D = 961 > 0 $$) --- ### **17. $$ \frac{1}{2}x^{2} + 2x + 3 = 0 $$** - **Coefficients:** $$ a = \frac{1}{2} $$, $$ b = 2 $$, $$ c = 3 $$ - **Discriminant:** $$ D = 2^2 - 4\left(\frac{1}{2}\right)(3) = 4 - 6 = -2 $$ - **Number of Real Solutions:** **No real solutions** (since $$ D = -2 < 0 $$) --- ### **18. $$ \frac{1}{6}x^{2} = 2x + 18 $$** - **Rearranged Equation:** $$ \frac{1}{6}x^{2} - 2x - 18 = 0 $$ - **Simplified Equation:** Multiply all terms by 6 to eliminate the fraction: $$ x^{2} - 12x - 108 = 0 $$ - **Coefficients:** $$ a = 1 $$, $$ b = -12 $$, $$ c = -108 $$ - **Discriminant:** $$ D = (-12)^2 - 4(1)(-108) = 144 + 432 = 576 $$ - **Number of Real Solutions:** **2** (since $$ D = 576 > 0 $$) --- **Summary Table:** | Problem | Equation | Discriminant $$ D $$ | Number of Real Solutions | |---------|--------------------------------|----------------------|-------------------------------| | 7 | $$ x^{2} + x - 42 = 0 $$ | 169 | 2 | | 8 | $$ -x^{2} + 13x - 40 = 0 $$ | 9 | 2 | | 9 | $$ x^{2} + 2x + 5 = 0 $$ | -16 | No real solutions | | 10 | $$ x^{2} - 18x + 81 = 0 $$ | 0 | 1 | | 11 | $$ -x^{2} + 7x + 44 = 0 $$ | 225 | 2 | | 12 | $$ \frac{1}{4}x^{2} - 5x +25 = 0 $$ | 0 | 1 | | 13 | $$ 2x^{2} - 5x + 7 = 0 $$ | -31 | No real solutions | | 14 | $$ 4x^{2} + 25x - 21 = 0 $$ | 961 | 2 | | 17 | $$ \frac{1}{2}x^{2} + 2x + 3 = 0 $$ | -2 | No real solutions | | 18 | $$ x^{2} - 12x - 108 = 0 $$ | 576 | 2 | --- If you need further explanations or assistance with solving for the actual roots, feel free to ask! Here are the discriminants and the number of real solutions for each equation: 1. **7. $$ x^{2} + x - 42 = 0 $$** - Discriminant: 169 - Real Solutions: 2 2. **8. $$ -x^{2} + 13x - 40 = 0 $$** - Discriminant: 9 - Real Solutions: 2 3. **9. $$ x^{2} + 2x + 5 = 0 $$** - Discriminant: -16 - Real Solutions: 0 4. **10. $$ x^{2} - 18x + 81 = 0 $$** - Discriminant: 0 - Real Solutions: 1 5. **11. $$ -x^{2} + 7x + 44 = 0 $$** - Discriminant: 225 - Real Solutions: 2 6. **12. $$ \frac{1}{4}x^{2} - 5x + 25 = 0 $$** - Discriminant: 0 - Real Solutions: 1 7. **13. $$ 2x^{2} - 5x + 7 = 0 $$** - Discriminant: -31 - Real Solutions: 0 8. **14. $$ 4x^{2} + 25x - 21 = 0 $$** - Discriminant: 961 - Real Solutions: 2 9. **17. $$ \frac{1}{2}x^{2} + 2x + 3 = 0 $$** - Discriminant: -2 - Real Solutions: 0 10. **18. $$ x^{2} - 12x - 108 = 0 $$** - Discriminant: 576 - Real Solutions: 2

Exercises
What is the value of the discriminant and what is the number of real solutions
for each equation?

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Related Questions

Latest Algebra Questions

Problem	Discriminant	Number of Real Solutions
7	169	2
8	9	2
9	-16	No real solutions
10	0	1
11	225	2
12	0	1
13	-31	No real solutions
14	961	2
17	-2	No real solutions
18	576	2

Exercises What is the value of the discriminant and what is the number of real solutions for each equation? Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.