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

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!