For questions 56: Find the discriminant (b^2-4ac) of the following quadratics. You can earn: 1 point for showing your work (what did you identify as a , b, and c; how did you substitute them) 1 point for working out the correct value. 5. 3x^2-4x+7=0 6 2x^2+5x-8=7

To find the discriminant of a quadratic equation in the form \( ax^2 + bx + c = 0 \), we use the formula for the discriminant: <br /><br />\[ \Delta = b^2 - 4ac \]<br /><br />Let's solve each problem step by step:<br /><br />### Problem 1: \( 3x^2 - 4x + 7 = 0 \)<br /><br />1. **Identify \( a \), \( b \), and \( c \):**<br /> - \( a = 3 \)<br /> - \( b = -4 \)<br /> - \( c = 7 \)<br /><br />2. **Substitute into the discriminant formula:**<br /><br /> \[<br /> \Delta = (-4)^2 - 4 \times 3 \times 7<br /> \]<br /><br />3. **Calculate:**<br /><br /> \[<br /> \Delta = 16 - 84 = -68<br /> \]<br /><br />The discriminant for the quadratic \( 3x^2 - 4x + 7 = 0 \) is \(-68\).<br /><br />### Problem 2: \( 2x^2 + 5x - 8 = 7 \)<br /><br />First, rewrite the equation in standard form by moving all terms to one side:<br /><br />\[ 2x^2 + 5x - 8 - 7 = 0 \]<br /><br />This simplifies to:<br /><br />\[ 2x^2 + 5x - 15 = 0 \]<br /><br />1. **Identify \( a \), \( b \), and \( c \):**<br /> - \( a = 2 \)<br /> - \( b = 5 \)<br /> - \( c = -15 \)<br /><br />2. **Substitute into the discriminant formula:**<br /><br /> \[<br /> \Delta = 5^2 - 4 \times 2 \times (-15)<br /> \]<br /><br />3. **Calculate:**<br /><br /> \[<br /> \Delta = 25 + 120 = 145<br /> \]<br /><br />The discriminant for the quadratic \( 2x^2 + 5x - 15 = 0 \) is \( 145 \).<br /><br />In summary:<br />- The discriminant for \( 3x^2 - 4x + 7 = 0 \) is \(-68\).<br />- The discriminant for \( 2x^2 + 5x - 15 = 0 \) is \( 145 \).