13. Q13
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Similar Question 1
<p>Determine the values of <code class='latex inline'>k</code> for which the function <code class='latex inline'>f(x) = 4x^2 - 3x + 2kx + 1</code> has two zeros. Check these value in the original equation.</p>
Similar Question 2
<p>For what value(s) of <code class='latex inline'>k</code> does the equation <code class='latex inline'>y=5x^2+6x+k</code> have each number of roots?</p><p><em>a)</em> two real roots</p><p><em>b)</em> one real root</p><p><em>c)</em> no real roots</p>
Similar Question 3
<p>If <code class='latex inline'>f(x) = x ^2 - 6x + 14</code> and <code class='latex inline'>g(x) = -x^2 - 20x - k</code>, determine the value <code class='latex inline'>k</code> so that there is exactly one point of intersection between the two parabolas. </p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Determine whether the vertex of each parabola lies above, below, or on the x-axis. Explain how you know,</p><p> <code class='latex inline'>h=5t^2-30t+45</code></p>
<p>For what value(s) of <code class='latex inline'>k</code> does the equation <code class='latex inline'>y=5x^2+6x+k</code> have each number of roots?</p><p><em>a)</em> two real roots</p><p><em>b)</em> one real root</p><p><em>c)</em> no real roots</p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 4x^2 - 2x </code></p>
<p>Factor each quadratic function to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 4x^2 - 1 </code></p>
<p>Determine the values of <code class='latex inline'>k</code> for which the function <code class='latex inline'>f(x) = 4x^2 - 3x + 2kx + 1</code> has two zeros. Check these value in the original equation.</p>
<p>Calculate the value of <code class='latex inline'>b^2 - 4ac</code> to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 9x^2 - 14.4x + 5.76 </code></p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 3x^2 + 2x + 7 </code></p>
<p>Determine the vertex and the direction of opening for each quadratic function. Then state the number of zeros.</p><p><code class='latex inline'> \displaystyle f(x) = -4x^2 + 7 </code></p>
<p>Calculate the value of <code class='latex inline'>b^2 - 4ac</code> to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = x^2 + 8x + 16 </code></p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p><code class='latex inline'>-x^2+8x=12</code></p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = -2.1x^2 + 9.06x -5.4 </code></p>
<p>Is it possible for <code class='latex inline'>n^2 + 25</code> to equal <code class='latex inline'>-8n</code>? Explain.</p>
<p>Determine the vertex and the direction of opening for each quadratic function. Then state the number of zeros.</p><p><code class='latex inline'> \displaystyle f(x) = 3x^2 - 5 </code></p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p><code class='latex inline'>x^2+3x-5=0</code></p>
<p>Factor the quadratic function to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 9x^2 + 6x + 1 </code></p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p> <code class='latex inline'>-17x-9=4x^2-5x</code></p>
<p>In the relation <code class='latex inline'>y=4x^2+24x-5</code>, for which values of <code class='latex inline'>y</code> will the corresponding equation have no solutions?</p>
<p>Use the discriminant to determine the number of roots for each equation.</p><p><code class='latex inline'>9x^2 -12x + 4 = 0 </code></p>
<p>State the number of times that each relation passes through the x-axis. Justify your answer</p><p><code class='latex inline'>y=-4.9x^2+5</code></p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p> <code class='latex inline'>6x^2+5x+12=0</code></p>
<p>Which quadratic equation has no real solutions?</p><p>A. <code class='latex inline'>\displaystyle 2 x^{2}+5 x-8=0 </code></p><p>B. <code class='latex inline'>\displaystyle -3 x^{2}+2 x-5=0 </code></p><p>C. <code class='latex inline'>\displaystyle -x^{2}+2 x+5=0 </code></p><p>D. <code class='latex inline'>\displaystyle 2 x^{2}+8 x+3=0 </code></p>
<p>Without graphing, determine the number of zeros that each relation has.</p><p><code class='latex inline'>y=3x^2+6x-8</code></p>
<p>Without graphing, determine how many <code class='latex inline'>\displaystyle x </code> -intercepts each function has.</p><p><code class='latex inline'>\displaystyle y=x^{2}+3 x+5 </code></p>
<p>If <code class='latex inline'>f(x) = x ^2 - 6x + 14</code> and <code class='latex inline'>g(x) = -x^2 - 20x - k</code>, determine the value <code class='latex inline'>k</code> so that there is exactly one point of intersection between the two parabolas. </p>
<p><code class='latex inline'>x^2-6x+5=0</code> has two roots <code class='latex inline'>x = 1, 5</code>.</p><p><em>i)</em> What does above tell you about the graph of <code class='latex inline'>y=x^2-6x+5</code>?</p><p><em>ii)</em> Verify your answer for part ii) using the discriminant.</p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 2x^2 -6x - 7 </code></p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = -2.4x^2 + x -1.2 </code></p>
<p>State the number of times that each relation passes through the x-axis. Justify your answer</p><p> <code class='latex inline'>y=3(x+2)^2-5</code></p>
<p>For each profit function, determine whether the company can break even. If the company can break even, determine in how many ways its can do so.</p><p><code class='latex inline'> \displaystyle P(x) = -0.3x^2 + 2x - 7.8 </code></p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p><code class='latex inline'>3x^2+2x=5x+12</code></p>
<p>Evaluate each expression for the given values of the variables.</p><p><code class='latex inline'>\displaystyle b^{2}-4 a c ; a=-5, b=2, c=4 </code></p>
<p>Factor each quadratic function to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 2x^2 - 6x </code></p>
<p>Show that <code class='latex inline'>(x^2 - 1)k = (x - 1)^2</code> has one solution for only one value of <code class='latex inline'>k</code>. </p>
<p>Use the discriminant to determine the number of real solutions that each equation has.</p><p><code class='latex inline'>-2x^2+8x-8=0</code></p>
<p>Without graphing, determine how many <code class='latex inline'>\displaystyle x </code> -intercepts each function has.</p><p><code class='latex inline'>\displaystyle y=3 x^{2}-10 x+6 </code></p>
<p>Use the discriminant to determine the number of roots for each equation.</p><p><code class='latex inline'>3x^2+ 4x - 5 = 0</code></p>
<p>A tangent is a line that touches a circle at exactly one point. For what values of <code class='latex inline'>k</code> will the line <code class='latex inline'>y=x+k</code> be tangent to the circle <code class='latex inline'>x^2+y^2=25</code>?</p>
<p>Without graphing, determine how many <code class='latex inline'>\displaystyle x </code> -intercepts each function has.</p><p><code class='latex inline'>\displaystyle y=x^{2}+17 x-2 </code></p>
<p>The height, <code class='latex inline'>h(t)</code>, of a projectile, in metres, can be modelled by the equation <code class='latex inline'>h(t) = 14t - 5t^2</code>, where t is the time in seconds after the projectile is released. Can the projectile ever reach a height of 9m? Explain.</p>
<p><strong>(a)</strong> Copy the graph of <code class='latex inline'>f(x) = (x - 2)^2 - 3</code>. Then draw lines with slope -4 that intersect the parabola at </p> <ul> <li>(i) one point,</li> <li>(ii) two points and </li> <li>(iii) no points.</li> </ul> <p><strong>(b)</strong> Write the equations of the lines from part (a).</p><p><strong>(c)</strong> How are all of the lines with slope - 4 that do not intersect the parabola related?</p>
<p>The graph of the function <code class='latex inline'>f(x) = x^2 - kx + k + 8</code> touches the <code class='latex inline'>x</code>-axis at one point. What are the possible values of <code class='latex inline'>k</code>?</p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = 3x^2 - x + 5 </code></p>
<p>Use the discriminant to determine the number of roots for each equation.</p><p><code class='latex inline'>-2x^2+ 5x - 1= 0</code></p>
<p>Factor each quadratic function to determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = x^2 - 6x - 16 </code></p>
<p>Without graphing, determine how many <code class='latex inline'>\displaystyle x </code> -intercepts each function has.</p><p><code class='latex inline'>\displaystyle y=7 x^{2}-2 x+9 </code></p>
<p>A parabola has equation <code class='latex inline'>y=4x^2+x+c</code>. One <code class='latex inline'>x</code>-intercept is —3. What is the value of <code class='latex inline'>c</code>?</p>
<p>Use the discriminant to determine the number of roots for each quadratic equation.</p><p><code class='latex inline'>3x^2 + 4x + \frac{4}{3}= 0</code></p>
<p>Which quadratic equation has exactly</p><p>one real root?</p><p>A. <code class='latex inline'>\displaystyle -9 x^{2}-6 x+1=0 </code></p><p>B. <code class='latex inline'>\displaystyle 9 x^{2}-6 x-1=0 </code></p><p>C. <code class='latex inline'>\displaystyle 9 x^{2}-6 x+1=0 </code></p><p>D. <code class='latex inline'>\displaystyle 9 x^{2}+6 x-1=0 </code></p>
<p>Use the discriminant to determine the number of roots for each quadratic equation.</p><p><code class='latex inline'>2x^2 - 8x + 9= 0</code></p>
<p>Determine the number of zeros. </p><p><code class='latex inline'> \displaystyle f(x) = -2x^2 + 6.4x -5.12 </code></p>
<p>Use the discriminant to determine the number of roots for each quadratic equation.</p><p><code class='latex inline'>x^2 - 5x +4 = 0</code></p>
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