Question 7 (2 points) Use discriminants to determine which of the following functions have at least one real root. Check all that apply. f(x)=-3x^2+x-2 b f(x)=-x^2+4 C f(x)=2x^2-12x+18 d f(x)=x^2-3x+25 e f(x)=10x^2-4x f f(x)=-x^2+10x-14

To determine which of the given functions have at least one real root, we need to calculate the discriminant for each function. The discriminant is given by the formula:<br /><br />\[ \Delta = b^2 - 4ac \]<br /><br />where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c \).<br /><br />Let's calculate the discriminant for each function:<br /><br />a) \( f(x) = -3x^2 + x - 2 \)<br />\[ \Delta = 1^2 - 4(-3)(-2) = 1 - 24 = -23 \]<br /><br />b) \( f(x) = -x^2 + 4 \)<br />\[ \Delta = 0^2 - 4(-1)(4) = 0 + 16 = 16 \]<br /><br />c) \( f(x) = 2x^2 - 12x + 18 \)<br />\[ \Delta = (-12)^2 - 4(2)(18) = 144 - 144 = 0 \]<br /><br />d) \( f(x) = x^2 - 3x + 25 \)<br />\[ \Delta = (-3)^2 - 4(1)(25) = 9 - 100 = -91 \]<br /><br />e) \( f(x) = 10x^2 - 4x \)<br />\[ \Delta = (-4)^2 - 4(10)(0) = 16 - 0 = 16 \]<br /><br />f) \( f(x) = -x^2 + 10x - 14 \)<br />\[ \Delta = 10^2 - 4(-1)(-14) = 100 - 56 = 44 \]<br /><br />Now, let's determine which functions have at least one real root based on the discriminant values:<br /><br />- If \( \Delta > 0 \), the quadratic equation has two distinct real roots.<br />- If \( \Delta = 0 \), the quadratic equation has one real root (a repeated root).<br />- If \( \Delta < 0 \), the quadratic equation has no real roots.<br /><br />Based on the discriminant values, the functions that have at least one real root are:<br /><br />b) \( f(x) = -x^2 + 4 \)<br />c) \( f(x) = 2x^2 - 12x + 18 \)<br />e) \( f(x) = 10x^2 - 4x \)<br />f) \( f(x) = -x^2 + 10x - 14 \)<br /><br />Therefore, the correct answers are:<br />b) \( f(x) = -x^2 + 4 \)<br />c) \( f(x) = 2x^2 - 12x + 18 \)<br />e) \( f(x) = 10x^2 - 4x \)<br />f) \( f(x) = -x^2 + 10x - 14 \)