The number \(b^{2} - 4ac\) tells you how many real roots a quadratic has without solving it. Put a line and a curve together, reduce them to one quadratic, and the same number tells you whether the line crosses the curve, touches it or misses it.
For \(ax^{2} + bx + c = 0\): \(b^{2} - 4ac > 0\) gives two distinct real roots, \(b^{2} - 4ac = 0\) two equal roots, \(b^{2} - 4ac < 0\) no real roots. For a line and a curve, substitute the line into the curve, rearrange to one quadratic \(= 0\) and use its discriminant: \(> 0\) the line meets the curve at two points, \(= 0\) the line is a tangent, \(< 0\) the line does not meet the curve.
Fig. 1: the curve y = x² + 1 with three lines. Substituting each line gives one quadratic; its discriminant Δ is positive (two meeting points), zero (a tangent) or negative (no meeting point)
Section 01
Three cases from one number.
In the formula \(x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) the roots are real only if the number under the square root is not negative. That number, \(b^{2} - 4ac\), is the discriminant. Always rearrange to \(ax^{2} + bx + c = 0\) first and read \(a\), \(b\) and \(c\) with their signs.
\(b^{2} - 4ac > 0\): two distinct real roots; the graph of \(y = ax^{2} + bx + c\) crosses the \(x\)-axis at two points.
\(b^{2} - 4ac = 0\): two equal roots (one repeated root); the graph touches the \(x\)-axis at its vertex.
\(b^{2} - 4ac < 0\): no real roots; the graph lies entirely above or entirely below the \(x\)-axis.
Worked example Count the roots
State the number of real roots of (a) \(2x^{2} - 7x + 3 = 0\) (b) \(9x^{2} + 6x + 1 = 0\) (c) \(3x^{2} - 2x + 1 = 0\).
(a) \(a = 2,\ b = -7,\ c = 3\): \((-7)^{2} - 4(2)(3) = 49 - 24 = 25 > 0\), so two distinct real roots.
(b) \(a = 9,\ b = 6,\ c = 1\): \(36 - 36 = 0\), so two equal roots. In fact \(9x^{2} + 6x + 1 = (3x + 1)^{2}\), so the root is \(x = -\dfrac{1}{3}\).
(c) \(a = 3,\ b = -2,\ c = 1\): \(4 - 12 = -8 < 0\), so no real roots.
Answer: (a) two distinct (b) two equal, \(x = -\frac{1}{3}\) (c) none
Worked example Rearrange before you read a, b, c
Show that the equation \(x(x + 5) = 2x - 3\) has no real roots.
Expand and collect everything on one side: \(x^{2} + 5x - 2x + 3 = 0\), so \(x^{2} + 3x + 3 = 0\).
\(a = 1,\ b = 3,\ c = 3\): \(b^{2} - 4ac = 9 - 12 = -3\).
\(-3 < 0\), so there are no real roots.
Answer: \(b^{2} - 4ac = -3 < 0\): no real roots
Examiner trap
Reading \(a\), \(b\), \(c\) straight from \(x(x + 5) = 2x - 3\) gives nonsense. The equation must be in the form \(ax^{2} + bx + c = 0\) first, and a negative \(b\) is squared with brackets: \((-7)^{2} = 49\), never \(-49\).
Section 02
Finding k: equations and inequalities.
When the quadratic contains an unknown constant \(k\), turn the wording into a condition on \(b^{2} - 4ac\). 'Equal roots' gives an equation in \(k\). 'Two distinct roots' and 'no real roots' give a quadratic inequality in \(k\), which you solve exactly as in the previous lesson: find the critical values, then decide whether you want the middle or the two outside parts.
Equal roots: \(b^{2} - 4ac = 0\).
Two distinct real roots: \(b^{2} - 4ac > 0\).
Real roots (equal allowed): \(b^{2} - 4ac \ge 0\).
No real roots: \(b^{2} - 4ac < 0\).
Worked example Equal roots
Find the values of \(k\) for which \(x^{2} + kx + k + 3 = 0\) has equal roots, and find the root in each case.
\(a = 1,\ b = k,\ c = k + 3\). Equal roots: \(k^{2} - 4(k + 3) = 0\).
Factorise: \((k - 1)(k - 9) < 0\). The critical values are \(1\) and \(9\).
The graph of \(y = (k - 1)(k - 9)\) is a U shape, below the axis between its roots.
Answer: \(1 < k < 9\)
Your turn
Find the set of values of \(k\) for which \(3x^{2} + kx + 12 = 0\) has two distinct real roots.
\(k^{2} - 4(3)(12) > 0\), so \(k^{2} > 144\), giving \(k < -12\) or \(k > 12\).
Examiner trap
Write the answer in the right form. 'Less than zero' between the critical values is ONE statement, \(1 < k < 9\). 'Greater than zero' is TWO separate pieces, \(k < -12\) or \(k > 12\); never write \(-12 > k > 12\), and never just \(k > 12\) from \(k^{2} > 144\).
Section 03
Lines that cross, touch or miss.
To find where the line \(y = mx + c\) meets a curve such as \(y = ax^{2} + bx + d\), set the two expressions for \(y\) equal. Rearranging gives ONE quadratic equation in \(x\), and each real root is the \(x\)-coordinate of a meeting point. So the discriminant of that new quadratic tells you how the line and curve sit (Fig. 1).
\(b^{2} - 4ac > 0\): the line meets (intersects) the curve at two distinct points.
\(b^{2} - 4ac = 0\): the line is a tangent to the curve; it touches at one point, the point of contact.
\(b^{2} - 4ac < 0\): the line does not meet the curve.
'The line meets the curve' (at least one point): \(b^{2} - 4ac \ge 0\).
Worked example Tangent and point of contact
Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x^{2} - 3x + 7\), and find the coordinates of the point of contact.
Set equal: \(x^{2} - 3x + 7 = x + c\), so \(x^{2} - 4x + (7 - c) = 0\).
Tangent: \((-4)^{2} - 4(1)(7 - c) = 0\), so \(16 - 28 + 4c = 0\), giving \(c = 3\).
With \(c = 3\): \(x^{2} - 4x + 4 = (x - 2)^{2} = 0\), so \(x = 2\) (a repeated root, as a tangent must give).
\(y = x + 3 = 5\). Check on the curve: \(4 - 6 + 7 = 5\).
Answer: \(c = 3\); point of contact \((2,\ 5)\)
Worked example Finding the gradient m
The line \(y = mx + 3\) and the curve \(y = 2x^{2} - x + 5\) are given. (a) Find the values of \(m\) for which the line is a tangent to the curve, and the point of contact in each case. (b) Find the set of values of \(m\) for which the line does not meet the curve.
Set equal: \(2x^{2} - x + 5 = mx + 3\), so \(2x^{2} - (1 + m)x + 2 = 0\).
Show that the line \(y = 3x - 5\) does not meet the curve \(y = x^{2} + x - 3\).
\(x^{2} + x - 3 = 3x - 5\) gives \(x^{2} - 2x + 2 = 0\); \(b^{2} - 4ac = 4 - 8 = -4 < 0\), so there are no meeting points.
Examiner trap
Use the discriminant of the COMBINED equation, after you have moved every term to one side. The discriminant of the curve on its own, \(x^{2} - 3x + 7\), only tells you about the \(x\)-axis, not about the line. And keep the brackets when the line's constant moves across: \(7 - c\), not \(7 + c\).
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What is the discriminant of \(3x^{2} + 4x - 2\)?
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\(x^{2} + 8x + k = 0\) has no real roots. Which describes \(k\)?
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Substituting \(y = 4x - 3\) into \(y = x^{2} + 1\) gives \(x^{2} - 4x + 4 = 0\). What does this tell you?
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The line \(y = mx + c\) does not meet a quadratic curve. Which statement is correct?
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Find the values of \(k\) for which \(x^{2} + (k - 1)x + 4 = 0\) has equal roots.
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Find the set of values of \(k\) for which the line \(y = x + k\) meets the curve \(y = x^{2} - 5x + 12\) at two distinct points.
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Find the values of \(m\) for which the line \(y = mx - 4\) is a tangent to the curve \(y = x^{2} + 5\).
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Find the set of values of \(k\) for which the line \(y = kx + 1\) does not meet the curve \(y = x^{2} + 2x + 2\).
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