Prove/show always positive

A question is this type if and only if it requires showing that a quadratic expression is always positive (or always has real roots) for all values of x, typically via completing the square or discriminant argument.

13 questions · Moderate -0.5

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CAIE P1 2023 March Q1
4 marks Moderate -0.5
1 A line has equation \(y = 3 x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant. Show that the line and the curve meet for all values of \(k\).
Edexcel C1 2011 June Q7
6 marks Moderate -0.3
7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.
  1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
  2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
  3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.
OCR MEI C1 2008 January Q11
12 marks Moderate -0.8
11
  1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 2009 January Q12
11 marks Moderate -0.8
12
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
  2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.
OCR MEI C1 Q15
12 marks Moderate -0.3
15
  1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 Q7
10 marks Standard +0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d22f53f5-ba80-4065-a94b-2a9c92c20dfb-2_696_879_960_673} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).
  1. Find the exact coordinates of the points of intersection of this curve with the axes.
  2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
  3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).
OCR MEI C1 Q4
12 marks Moderate -0.3
4
  1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
  2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
  3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).
OCR MEI C1 2011 January Q13
10 marks Standard +0.8
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-4_686_878_936_632} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).
  1. Find the exact coordinates of the points of intersection of this curve with the axes.
  2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
  3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).
Edexcel AS Paper 1 2018 June Q2
5 marks Moderate -0.8
  1. (i) Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\) (ii) "If I add 3 to a number and square the sum, the result is greater than the square of the original number."
State, giving a reason, if the above statement is always true, sometimes true or never true.
Edexcel Paper 2 Specimen Q6
6 marks Moderate -0.8
6. Complete the table below. The first one has been done for you. For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case.
StatementAlways TrueSometimes TrueNever TrueReason
The quadratic equation \(a x ^ { 2 } + b x + c = 0 , \quad ( a \neq 0 )\) has 2 real roots.✓It only has 2 real roots when \(b ^ { 2 } - 4 a c > 0\). When \(b ^ { 2 } - 4 a c = 0\) it has 1 real root and when \(b ^ { 2 } - 4 a c < 0\) it has 0 real roots.
(i)
When a real value of \(x\) is substituted into \(x ^ { 2 } - 6 x + 10\) the result is positive.
(ii)
If \(a x > b\) then \(x > \frac { b } { a }\)
(2)
(iii)
The difference between consecutive square numbers is odd.
AQA C1 2009 January Q4
10 marks Easy -1.2
4
    1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
  2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
    1. Write down the coordinates of the minimum point of the curve.
    2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).
OCR MEI AS Paper 1 2019 June Q1
3 marks Easy -1.2
1 In this question you must show detailed reasoning. Show that the equation \(x = 7 + 2 x ^ { 2 }\) has no real roots.
SPS SPS SM 2025 October Q9
4 marks Moderate -0.8
  1. Show that the equation \(x^2 + kx - k^2 = 0\) has real roots for all real values of \(k\). [2]
  2. Show that the roots of the equation \(x^2 + kx - k^2 = 0\) are \(\left(\frac{-1 \pm \sqrt{5}}{2}\right)k\). [2]