Solve each by using the quadratic formula. 1 5x^2+21x=-18 2 81x^2=9 3 8x^2+12x=8 4 4x^2=-16x-16 5 10x^2=-7x+6 6 -3x^2=8x-12 7 2x^2=12x-18 8 -3x^2=12-12x 9 0.2x^2-1.5x+2.9=0 10 2x^2-5x=-20 State the discriminant for each equation. Then determine the number of real solutions of the equation. 11 0.2x^2-1.5x+2.9=0 12 2.25x^2-3x=-1 13 x^2-(4)/(5)x=3 14 0.5x^2-2x=-2 15 2x^2=(5)/(2)x+(3)/(2)

Let's solve each quadratic equation using the quadratic formula and then state the discriminant for each equation to determine the number of real solutions.<br /><br />### Solving Quadratic Equations Using the Quadratic Formula<br /><br />The quadratic formula is given by:<br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />where \( ax^2 + bx + c = 0 \).<br /><br />#### 1. \( 5x^2 + 21x = -18 \)<br /><br />Rewrite the equation in standard form:<br />\[ 5x^2 + 21x + 18 = 0 \]<br /><br />Here, \( a = 5 \), \( b = 21 \), and \( c = 18 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = 21^2 - 4(5)(18) = 441 - 360 = 81 \]<br /><br />Since the discriminant is positive, there are two real solutions.<br /><br />#### 2. \( 81x^2 = 9 \)<br /><br />Rewrite the equation in standard form:<br />\[ 81x^2 - 9 = 0 \]<br />\[ 81x^2 - 9 = 0 \]<br /><br />Here, \( a = 81 \), \( b = 0 \), and \( c = -9 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = 0^2 - 4(81)(-9) = 0 + 2916 = 2916 \]<br /><br />Since the discriminant is positive, there are two real solutions.<br /><br />#### 3. \( 8x^2 + 12x = 8 \)<br /><br />Rewrite the equation in standard form:<br />\[ 8x^2 + 12x - 8 = 0 \]<br /><br />Here, \( a = 8 \), \( b = 12 \), and \( c = -8 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = 12^2 - 4(8)(-8) = 144 + 256 = 400 \]<br /><br />Since the discriminant is positive, there are two real solutions.<br /><br />#### 4. \( 4x^2 = -16x - 16 \)<br /><br />Rewrite the equation in standard form:<br />\[ 4x^2 + 16x + 16 = 0 \]<br /><br />Here, \( a = 4 \), \( b = 16 \), and \( c = 16 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = 16^2 - 4(4)(16) = 256 - 256 = 0 \]<br /><br />Since the discriminant is zero, there is one real solution.<br /><br />#### 5. \( 10x^2 = -7x + 6 \)<br /><br />Rewrite the equation in standard form:<br />\[ 10x^2 + 7x - 6 = 0 \]<br /><br />Here, \( a = 10 \), \( b = 7 \), and \( c = -6 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = 7^2 - 4(10)(-6) = 49 + 240 = 289 \]<br /><br />Since the discriminant is positive, there are two real solutions.<br /><br />#### 6. \( -3x^2 = 8x - 12 \)<br /><br />Rewrite the equation in standard form:<br />\[ -3x^2 - 8x + 12 = 0 \]<br /><br />Here, \( a = -3 \), \( b = -8 \), and \( c = 12 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = (-8)^2 - 4(-3)(12) = 64 + 144 = 208 \]<br /><br />Since the discriminant is positive, there are two real solutions.<br /><br />#### 7. \( 2x^2 = 12x - 18 \)<br /><br />Rewrite the equation in standard form:<br />\[ 2x^2 - 12x + 18 = 0 \]<br /><br />Here, \( a = 2 \), \( b = -12 \), and \( c = 18 \).<br /><br />Calculate the discriminant:<br />\[ \Delta = b^2 - 4ac = (-12)^2 - 4(2)(18) = 144 - 144 = 0 \]<br /><br />Since the discriminant is zero, there is one real solution.<br /><br />#### 8. \( -3x^2 = 12 - 12x \)<br /><br />Rewrite the equation in standard form:<br />\[ -3x^2 + 12x - 12 = 0 \]<br /><br />Here, \( a = -3 \), \( b = 12 \), and \( c = -12 \).<br /><br />Calculate the discrimin