Quadratic Formula and the Discriminant Practice

Write the sign with each coefficient before using the quadratic formula.

Standard form is \(ax^2+bx+c=0\). Reading each signed coefficient gives \(a=3\), \(b=-7\), and \(c=2\).

Write the sign with each coefficient before using the quadratic formula.

Standard form is \(ax^2+bx+c=0\). Reading each signed coefficient gives \(a=-2\), \(b=5\), and \(c=-9\).

Write the sign with each coefficient before using the quadratic formula.

Standard form is \(ax^2+bx+c=0\). Reading each signed coefficient gives \(a=4\), \(b=0\), and \(c=-11\).

Write the sign with each coefficient before using the quadratic formula.

Standard form is \(ax^2+bx+c=0\). Reading each signed coefficient gives \(a=1\), \(b=-12\), and \(c=8\).

Write the sign with each coefficient before using the quadratic formula.

Standard form is \(ax^2+bx+c=0\). Reading each signed coefficient gives \(a=-5\), \(b=-3\), and \(c=6\).

Calculate the squared linear coefficient and signed product separately.

Use \(D=b^2-4ac\): \(D=(5)^2-4(1)(2)=17\).

Calculate the squared linear coefficient and signed product separately.

Use \(D=b^2-4ac\): \(D=(-6)^2-4(2)(1)=28\).

Calculate the squared linear coefficient and signed product separately.

Use \(D=b^2-4ac\): \(D=(4)^2-4(3)(-5)=76\).

Calculate the squared linear coefficient and signed product separately.

Use \(D=b^2-4ac\): \(D=(8)^2-4(-1)(3)=76\).

Calculate the squared linear coefficient and signed product separately.

Use \(D=b^2-4ac\): \(D=(-2)^2-4(4)(7)=-108\).

Connect the discriminant sign to the number of horizontal-axis intersections.

The discriminant is \(36\), which is positive. Therefore there are 2 real roots, matching the number of horizontal-axis intersections.

Connect the discriminant sign to the number of horizontal-axis intersections.

The discriminant is \(0\), which is zero. Therefore there is 1 real root, matching the number of horizontal-axis intersections.

Connect the discriminant sign to the number of horizontal-axis intersections.

The discriminant is \(-36\), which is negative. Therefore there are 0 real roots, matching the number of horizontal-axis intersections.

Connect the discriminant sign to the number of horizontal-axis intersections.

The discriminant is \(36\), which is positive. Therefore there are 2 real roots, matching the number of horizontal-axis intersections.

Connect the discriminant sign to the number of horizontal-axis intersections.

The discriminant is \(0\), which is zero. Therefore there is 1 real root, matching the number of horizontal-axis intersections.

Compute the discriminant first, then simplify the plus and minus branches separately.

Substituting the signed coefficients into \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) produces \(x=-4\) and \(x=7\). Both values make the original quadratic zero.

Compute the discriminant first, then simplify the plus and minus branches separately.

Substituting the signed coefficients into \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) produces \(x=-3\) and \(x=5\). Both values make the original quadratic zero.

Compute the discriminant first, then simplify the plus and minus branches separately.

Substituting the signed coefficients into \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) produces \(x=2\) and \(x=8\). Both values make the original quadratic zero.

Compute the discriminant first, then simplify the plus and minus branches separately.

Substituting the signed coefficients into \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) produces \(x=-2\) and \(x=4\). Both values make the original quadratic zero.

Compute the discriminant first, then simplify the plus and minus branches separately.

Substituting the signed coefficients into \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) produces \(x=1\) and \(x=6\). Both values make the original quadratic zero.

Keep the entire numerator over the denominator and reduce common factors after simplifying the radical.

Using \(a=1\), \(b=2\), and \(c=-5\), the discriminant is \(24\). Substitution and radical simplification give \(x=-1\pm\sqrt6\). Both branches satisfy the original equation.

Keep the entire numerator over the denominator and reduce common factors after simplifying the radical.

Using \(a=2\), \(b=4\), and \(c=-3\), the discriminant is \(40\). Substitution and radical simplification give \(x=-1\pm\frac{\sqrt{10}}2\). Both branches satisfy the original equation.

Keep the entire numerator over the denominator and reduce common factors after simplifying the radical.

Using \(a=3\), \(b=-6\), and \(c=-2\), the discriminant is \(60\). Substitution and radical simplification give \(x=1\pm\frac{\sqrt{15}}3\). Both branches satisfy the original equation.

Keep the entire numerator over the denominator and reduce common factors after simplifying the radical.

Using \(a=1\), \(b=-4\), and \(c=-1\), the discriminant is \(20\). Substitution and radical simplification give \(x=2\pm\sqrt5\). Both branches satisfy the original equation.

Keep the entire numerator over the denominator and reduce common factors after simplifying the radical.

Using \(a=2\), \(b=-2\), and \(c=-5\), the discriminant is \(44\). Substitution and radical simplification give \(x=\frac{1\pm\sqrt{11}}2\). Both branches satisfy the original equation.

Tangency corresponds to a zero discriminant; solve the resulting parameter equation completely.

Exactly one real solution requires \(D=0\): \(k^2-4(4)=0\), so \(k^2=16\) and \(k=\pm4\).

Tangency corresponds to a zero discriminant; solve the resulting parameter equation completely.

Exactly one real solution requires \(D=0\): \(k^2-4(9)=0\), so \(k^2=36\) and \(k=\pm6\).

Tangency corresponds to a zero discriminant; solve the resulting parameter equation completely.

Exactly one real solution requires \(D=0\): \(k^2-4(16)=0\), so \(k^2=64\) and \(k=\pm8\).

Tangency corresponds to a zero discriminant; solve the resulting parameter equation completely.

Exactly one real solution requires \(D=0\): \(k^2-4(25)=0\), so \(k^2=100\) and \(k=\pm10\).

Tangency corresponds to a zero discriminant; solve the resulting parameter equation completely.

Exactly one real solution requires \(D=0\): \(k^2-4(36)=0\), so \(k^2=144\) and \(k=\pm12\).

Use coefficients to calculate the discriminant, then use the graph as a consistency check.

Direct calculation gives \(D=(-4)^2-4(1)(3)=4\). Its sign predicts that the graph crosses the horizontal axis twice, exactly as rendered.

Use coefficients to calculate the discriminant, then use the graph as a consistency check.

Direct calculation gives \(D=(2)^2-4(1)(1)=0\). Its sign predicts that the graph touches the horizontal axis once, exactly as rendered.

Use coefficients to calculate the discriminant, then use the graph as a consistency check.

Direct calculation gives \(D=(0)^2-4(2)(6)=-48\). Its sign predicts that the graph does not meet the horizontal axis, exactly as rendered.

Use coefficients to calculate the discriminant, then use the graph as a consistency check.

Direct calculation gives \(D=(-2)^2-4(-1)(8)=36\). Its sign predicts that the graph crosses the horizontal axis twice, exactly as rendered.

Use coefficients to calculate the discriminant, then use the graph as a consistency check.

Direct calculation gives \(D=(6)^2-4(3)(5)=-24\). Its sign predicts that the graph does not meet the horizontal axis, exactly as rendered.

Use the root-sum relationship when individual roots are not requested.

For \(ax^2+bx+c=0\), the sum is \(-\frac ba=-\frac{-10}{2}=5\).

Use the root-sum relationship when individual roots are not requested.

For \(ax^2+bx+c=0\), the sum is \(-\frac ba=-\frac{12}{3}=-4\).

Use the root-sum relationship when individual roots are not requested.

For \(ax^2+bx+c=0\), the sum is \(-\frac ba=-\frac{14}{-2}=7\).

Use the root-sum relationship when individual roots are not requested.

For \(ax^2+bx+c=0\), the sum is \(-\frac ba=-\frac{-15}{5}=3\).

Use the root-sum relationship when individual roots are not requested.

For \(ax^2+bx+c=0\), the sum is \(-\frac ba=-\frac{-20}{-4}=-5\).

Match the requested relationship to the coefficient formula before using the full quadratic formula.

The product of roots is \(\frac ca=\frac{-12}{2}=-6\). Solving for each root is unnecessary.

Match the requested relationship to the coefficient formula before using the full quadratic formula.

The product of roots is \(\frac ca=\frac{20}{4}=5\). Solving for each root is unnecessary.

Match the requested relationship to the coefficient formula before using the full quadratic formula.

The product of roots is \(\frac ca=\frac{15}{-3}=-5\). Solving for each root is unnecessary.

Match the requested relationship to the coefficient formula before using the full quadratic formula.

The product of roots is \(\frac ca=\frac{-25}{5}=-5\). Solving for each root is unnecessary.

Match the requested relationship to the coefficient formula before using the full quadratic formula.

The product of roots is \(\frac ca=\frac{-14}{-2}=7\). Solving for each root is unnecessary.

Write signed coefficients and the discriminant on separate lines before substitution.

The linear coefficient is negative, so b must be recorded with a negative sign. Standard form, a separately computed discriminant, and substitution of final roots provide independent checks.

Write signed coefficients and the discriminant on separate lines before substitution.

The discriminant uses subtraction: b squared minus four times a times c. Standard form, a separately computed discriminant, and substitution of final roots provide independent checks.

Write signed coefficients and the discriminant on separate lines before substitution.

The entire numerator, including the opposite of b, is divided by 2a. Standard form, a separately computed discriminant, and substitution of final roots provide independent checks.

Write signed coefficients and the discriminant on separate lines before substitution.

A zero discriminant gives one repeated real root. Standard form, a separately computed discriminant, and substitution of final roots provide independent checks.

Write signed coefficients and the discriminant on separate lines before substitution.

Use negative b over a to answer directly and reduce arithmetic risk. Standard form, a separately computed discriminant, and substitution of final roots provide independent checks.